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<p><strong>Description:</strong> Builds upon previous lectures to cover additional graph classifications and criteria, including tournament graphs and directed acyclic graphs. Also covers Euler Tours, Hamiltonian paths, and adjacency matrices.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>
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<p><span m="390">The</span> <span m="510">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1670">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2430">Creative</span> <span m="2830">Commons</span> <span m="3230">license.</span> <span m="4350">Your</span> <span m="4530">support</span> <span m="5030">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high</span> <span m="7930">quality</span> <span m="8440">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation</span> <span m="12000">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13130">materials</span> <span m="13670">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14780">courses,</span> <span m="15900">visit</span> <span m="16120">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="23140">PROFESSOR: So</span> <span m="23520">today we're</span> <span m="23850">going</span> <span m="24110">to</span> <span m="24210">continue</span> <span m="24680">our</span> <span m="25220">course</span> <span m="25520">on</span> <span m="25700">the</span> <span m="25840">graph</span> <span m="26140">theory.</span> <span m="26960">It's</span> <span m="27120">going</span> <span m="27270">to</span> <span m="27340">be</span> <span m="27470">a</span> <span m="27670">mixture</span> <span m="27940">of</span> <span m="28050">all</span> <span m="28220">kinds</span> <span m="28500">of</span> <span m="28600">topics.</span> <span m="30110">We</span> <span m="30220">first</span> <span m="30510">start</span> <span m="30780">off</span> <span m="31020">with</span> <span m="31550">Euler</span> <span m="31830">tour,</span> <span m="32729">and</span> <span m="32840">then</span> <span m="32980">we</span> <span m="33060">get</span> <span m="33250">into</span> <span m="33570">directed</span> <span m="34030">graphs</span> <span m="34810">and</span> <span m="35400">cover</span> <span m="35710">the</span> <span m="35900">definitions.</span> <span m="37500">And</span> <span m="37920">we'll</span> <span m="38060">talk</span> <span m="38270">about</span> <span m="38450">a</span> <span m="38500">special</span> <span m="38900">type,</span> <span m="39630">which</span> <span m="39850">are</span> <span m="39940">called</span> <span m="40270">tournament</span> <span m="40650">graphs.</span> <span m="42030">And</span> <span m="42560">we'll</span> <span m="42710">do</span> <span m="43900">a</span> <span m="43960">whole</span> <span m="44160">bunch</span> <span m="44360">of</span> <span m="44460">proofs,</span> <span m="45040">and</span> <span m="45410">hopefully</span> <span m="46940">you</span> <span m="47060">will all</span> <span m="47310">contribute</span> <span m="47880">to</span> <span m="48380">make</span> <span m="49460">this</span> <span m="49640">work</span> <span m="50220">and</span> <span m="50680">think</span> <span m="50950">about</span> <span m="51250">how</span> <span m="51410">to</span> <span m="51530">do</span> <span m="51680">this.</span> </p>
<p><span m="52750">So</span> <span m="53510">next</span> <span m="53820">week</span> <span m="54000">we</span> <span m="54130">will</span> <span m="54530">continue</span> <span m="54880">with</span> <span m="55230">graph</span> <span m="55520">theory,</span> <span m="56110">and</span> <span m="56330">we</span> <span m="56470">will</span> <span m="56760">discuss</span> <span m="57170">a</span> <span m="57210">very</span> <span m="57440">special</span> <span m="57870">type.</span> <span m="59940">We</span> <span m="60080">will</span> <span m="60170">use</span> <span m="60660">directed</span> <span m="61090">graphs</span> <span m="62430">in</span> <span m="62610">communication</span> <span m="63230">networks.</span> <span m="63970">And</span> <span m="65590">on</span> <span m="65700">Thursday,</span> <span m="66230">we'll</span> <span m="66460">actually</span> <span m="67050">use</span> <span m="68150">these</span> <span m="68430">special</span> <span m="68800">types</span> <span m="69070">of</span> <span m="69150">graphs</span> <span m="69470">that</span> <span m="69500">we'll</span> <span m="69740">talk</span> <span m="69990">about</span> <span m="70240">in</span> <span m="70340">a</span> <span m="70400">moment,</span> <span m="71160">DAGs.</span> </p>
<p><span m="73960">So</span> <span m="74170">let's</span> <span m="74350">talk</span> <span m="74530">about</span> <span m="74990">Euler</span> <span m="75300">tour.</span> <span m="77250">Euler,</span> <span m="78160">a</span> <span m="78490">famous</span> <span m="78920">mathematician,</span> <span m="79810">he</span> <span m="80510">asked</span> <span m="80690">the</span> <span m="80790">question</span> <span m="81280">like--</span> <span m="82336">he</span> <span m="82690">lived</span> <span m="83000">in</span> <span m="83310">Konigsberg.</span> <span m="84865">And</span> <span m="85150">there</span> <span m="85320">were</span> <span m="85410">seven</span> <span m="85750">bridges.</span> <span m="86180">He</span> <span m="86310">was</span> <span m="86440">wondering,</span> <span m="86810">can</span> <span m="86960">you</span> <span m="87050">traverse</span> <span m="87700">all</span> <span m="87970">the</span> <span m="88050">bridges</span> <span m="89940">exactly</span> <span m="90540">once.</span> <span m="90810">So</span> <span m="91170">he</span> <span m="91310">would</span> <span m="91420">start</span> <span m="91710">walking</span> <span m="92170">and</span> <span m="92270">try</span> <span m="92500">to</span> <span m="92710">cross</span> <span m="93040">all</span> <span m="93300">those</span> <span m="93540">bridges.</span> <span m="93980">And</span> <span m="94470">apparently</span> <span m="95320">this</span> <span m="95540">little</span> <span m="95780">islands</span> <span m="96150">in</span> <span m="96530">sort</span> <span m="96800">of</span> <span m="96910">the</span> <span m="97000">river</span> <span m="97310">as</span> <span m="97450">well,</span> <span m="97700">and</span> <span m="97840">so</span> <span m="98040">on.</span> <span m="98620">So</span> <span m="98800">how</span> <span m="98950">can</span> <span m="99080">you</span> <span m="99160">do</span> <span m="99310">that?</span> </p>
<p><span m="100980">This</span> <span m="101170">is</span> <span m="101310">actually</span> <span m="101560">the</span> <span m="101650">birth</span> <span m="101980">of</span> <span m="102090">graph</span> <span m="102420">theory.</span> <span m="103390">And</span> <span m="104600">this</span> <span m="104740">particular</span> <span m="105120">problem</span> <span m="105570">is</span> <span m="105720">named</span> <span m="105960">after</span> <span m="106220">him.</span> <span m="107030">So</span> <span m="107170">what</span> <span m="107290">is</span> <span m="107450">an</span> <span m="107660">Euler</span> <span m="108130">tour?</span> <span m="112830">It's</span> <span m="113080">actually</span> <span m="113360">defined</span> <span m="115260">as</span> <span m="115690">a</span> <span m="115900">special</span> <span m="119550">walk.</span> <span m="126380">It's</span> <span m="126810">a</span> <span m="126870">walk</span> <span m="127210">that</span> <span m="127720">traverses</span> <span m="130259">every</span> <span m="130419">edge</span> <span m="133400">exactly</span> <span m="133850">once.</span> </p>
<p><span m="143110">So</span> <span m="143330">it</span> <span m="143420">turns</span> <span m="143700">out</span> <span m="143910">the</span> <span m="144020">you</span> <span m="144190">can</span> <span m="144350">actually</span> <span m="144960">characterize</span> <span m="145640">these</span> <span m="145970">types</span> <span m="146290">of</span> <span m="146400">graphs.</span> <span m="148210">So</span> <span m="148860">we</span> <span m="148950">are</span> <span m="149020">talking</span> <span m="149340">here</span> <span m="149680">about</span> <span m="150390">undirected</span> <span m="150980">graph,</span> <span m="151330">so</span> <span m="152650">continuation</span> <span m="153370">of</span> <span m="153790">last</span> <span m="154150">time.</span> <span m="154990">So</span> <span m="155390">the</span> <span m="155580">edges</span> <span m="155840">that</span> <span m="156070">we</span> <span m="156160">consider</span> <span m="156700">have</span> <span m="156880">no</span> <span m="157070">direction.</span> <span m="157690">We'll</span> <span m="157880">come</span> <span m="158030">back</span> <span m="158300">to</span> <span m="158410">that</span> <span m="159260">in</span> <span m="159440">a</span> <span m="159490">moment.</span> <span m="159840">We will</span> <span m="160140">start</span> <span m="160390">defining</span> <span m="160780">those</span> <span m="161030">later.</span> <span m="162210">So</span> <span m="162770">Euler</span> <span m="163100">tour</span> <span m="163280">is</span> <span m="163400">a</span> <span m="163510">walk</span> <span m="163760">that</span> <span m="163960">traverses</span> <span m="164270">every</span> <span m="164380">edge</span> <span m="164660">exactly</span> <span m="165220">once.</span> <span m="166240">And</span> <span m="168090">at</span> <span m="168200">the</span> <span m="168240">same</span> <span m="168520">time,</span> <span m="169630">it's</span> <span m="169900">also</span> <span m="170140">a</span> <span m="170280">tour.</span> <span m="170840">So</span> <span m="171370">that</span> <span m="171480">means</span> <span m="171790">that</span> <span m="171910">it</span> <span m="172150">actually</span> <span m="172520">starts</span> <span m="174910">and</span> <span m="175120">finishes</span> <span m="178700">at</span> <span m="179420">the</span> <span m="179490">same</span> <span m="179720">vertex.</span> </p>
<p><span m="190050">Now</span> <span m="190180">it</span> <span m="190240">turns</span> <span m="190490">out</span> <span m="190840">that</span> <span m="191350">undirected</span> <span m="192210">graphs,</span> <span m="192485">the</span> <span m="192760">graphs</span> <span m="193500">that</span> <span m="193640">we've</span> <span m="193810">been</span> <span m="193930">talking</span> <span m="194260">about,</span> <span m="196530">those</span> <span m="196800">that</span> <span m="196890">Euler</span> <span m="197210">tours</span> <span m="198090">can</span> <span m="198330">be</span> <span m="198510">easily</span> <span m="198820">characterized.</span> <span m="199680">And</span> <span m="199810">that's</span> <span m="199970">the</span> <span m="200060">theorem</span> <span m="200560">that</span> <span m="200610">we're going</span> <span m="200810">to</span> <span m="200910">prove</span> <span m="201480">next.</span> <span m="203460">The</span> <span m="203540">theorem</span> <span m="203870">states</span> <span m="204520">that</span> <span m="206620">actually,</span> <span m="208390">if</span> <span m="208600">a</span> <span m="208730">connected</span> <span m="209220">graph</span> <span m="214990">has</span> <span m="216880">an</span> <span m="217230">Euler tour,</span> <span m="222590">if</span> <span m="222800">and</span> <span m="222930">only</span> <span m="223180">if,</span> <span m="228460">every</span> <span m="229000">vertex</span> <span m="230050">of the</span> <span m="230300">graph</span> <span m="232260">has</span> <span m="232470">even</span> <span m="232800">degree.</span> <span m="236120">So</span> <span m="236350">that's</span> <span m="236730">a</span> <span m="237800">very</span> <span m="238100">nice</span> <span m="238550">and</span> <span m="238800">simple</span> <span m="239130">and</span> <span m="239220">straightforward</span> <span m="239730">characterization.</span> </p>
<p><span m="240560">So</span> <span m="240720">how</span> <span m="240910">can we</span> <span m="241325">prove</span> <span m="241740">this?</span> <span m="244980">So</span> <span m="245230">let's</span> <span m="245420">have</span> <span m="245550">a</span> <span m="245650">look.</span> <span m="248820">So</span> <span m="249000">first</span> <span m="249300">of</span> <span m="249340">all,</span> <span m="249990">we</span> <span m="250180">have</span> <span m="250880">an</span> <span m="251090">if</span> <span m="251470">and</span> <span m="251890">only</span> <span m="252220">if</span> <span m="252350">statement.</span> <span m="254140">So</span> <span m="255120">we</span> <span m="255220">need</span> <span m="255390">to</span> <span m="255520">prove</span> <span m="255980">two</span> <span m="256220">directions.</span> <span m="256930">We</span> <span m="256990">need</span> <span m="257149">to</span> <span m="257240">proof</span> <span m="257589">that</span> <span m="258550">if</span> <span m="258810">I</span> <span m="258899">have</span> <span m="259140">a</span> <span m="259250">connected</span> <span m="259649">graph</span> <span m="260070">that</span> <span m="260200">has</span> <span m="260440">an</span> <span m="260649">Euler</span> <span m="260979">tour,</span> <span m="261310">I</span> <span m="261430">need</span> <span m="261589">to</span> <span m="261680">show</span> <span m="261990">that</span> <span m="262730">every</span> <span m="262950">vertex</span> <span m="263430">has</span> <span m="263550">even</span> <span m="263810">degree.</span> <span m="264470">And</span> <span m="264660">also</span> <span m="264990">the</span> <span m="265090">reverse--</span> <span m="266100">if</span> <span m="266320">I</span> <span m="266400">have</span> <span m="266530">a</span> <span m="266560">graph</span> <span m="266910">for</span> <span m="267410">which</span> <span m="267570">every</span> <span m="267860">vertex has</span> <span m="268140">even</span> <span m="268370">degree,</span> <span m="268710">I</span> <span m="268790">need</span> <span m="268960">to</span> <span m="269040">show</span> <span m="269320">that</span> <span m="269420">it has</span> <span m="269530">an</span> <span m="269850">Euler</span> <span m="270140">tour.</span> <span m="271080">So</span> <span m="271250">the</span> <span m="271370">proof</span> <span m="272470">consists</span> <span m="272830">of</span> <span m="272940">two</span> <span m="273140">parts.</span> </p>
<p><span m="274590">So</span> <span m="274650">let's</span> <span m="274830">first</span> <span m="276060">do</span> <span m="276180">this</span> <span m="276400">implication,</span> <span m="278430">where</span> <span m="278680">we</span> <span m="278760">assure</span> <span m="279480">that</span> <span m="279660">we</span> <span m="279760">have</span> <span m="279890">connected</span> <span m="280280">graph</span> <span m="280570">that</span> <span m="280650">has</span> <span m="280940">an</span> <span m="281250">Euler</span> <span m="281460">tour.</span> <span m="282660">So</span> <span m="283650">assume</span> <span m="284190">we</span> <span m="284290">have</span> <span m="284480">a</span> <span m="284520">graph.</span> <span m="287030">So</span> <span m="287310">we</span> <span m="287440">have</span> <span m="287680">G</span> <span m="288380">with</span> <span m="288550">the</span> <span m="288620">vertex</span> <span m="288860">at</span> <span m="289240">V,</span> <span m="289660">edge</span> <span m="290000">is</span> <span m="290170">at</span> <span m="290390">E,</span> <span m="292370">and</span> <span m="292800">it</span> <span m="292950">has</span> <span m="293230">an</span> <span m="293420">Euler</span> <span m="293730">tour.</span> </p>
<p><span m="295810">So</span> <span m="296040">what</span> <span m="296090">does</span> <span m="296380">it</span> <span m="296400">mean?</span> <span m="297730">Well,</span> <span m="298820">it's</span> <span m="299010">really</span> <span m="299250">means</span> <span m="300450">that</span> <span m="300910">we</span> <span m="301190">can</span> <span m="301380">walk</span> <span m="301810">from</span> <span m="302090">some</span> <span m="302860">start</span> <span m="303710">vertex,</span> <span m="304300">V0.</span> <span m="305960">We</span> <span m="306080">can</span> <span m="306250">go</span> <span m="306460">all</span> <span m="306690">the</span> <span m="306770">way</span> <span m="306950">around</span> <span m="307390">to</span> <span m="307620">V1,</span> <span m="308650">and</span> <span m="308940">some</span> <span m="309120">more,</span> <span m="310380">and</span> <span m="310620">so</span> <span m="310840">on,</span> <span m="311440">to</span> <span m="312470">V,</span> <span m="313120">say,</span> <span m="313370">k</span> <span m="313620">minus</span> <span m="314080">1.</span> <span m="315170">And</span> <span m="315410">then</span> <span m="315940">end</span> <span m="316180">vertex</span> <span m="317170">is</span> <span m="317380">the</span> <span m="317450">same</span> <span m="317770">as</span> <span m="317900">the</span> <span m="317990">start</span> <span m="318370">vertex.</span> <span m="319510">So</span> <span m="319890">we</span> <span m="320010">have</span> <span m="320940">a</span> <span m="321090">walk</span> <span m="321340">that</span> <span m="321570">goes</span> <span m="321860">all</span> <span m="322160">around</span> <span m="322980">the</span> <span m="323060">whole</span> <span m="323310">graph.</span> <span m="324060">So</span> <span m="324570">every</span> <span m="325040">edge</span> <span m="325760">in</span> <span m="326540">this</span> <span m="327110">walk</span> <span m="329270">are</span> <span m="329440">exactly</span> <span m="329910">the</span> <span m="330060">edges</span> <span m="330770">that</span> <span m="331070">are</span> <span m="331200">in the</span> <span m="331240">graph.</span> <span m="331740">And</span> <span m="332700">each</span> <span m="333090">edge</span> <span m="333350">in</span> <span m="333440">the</span> <span m="333510">graph</span> <span m="333780">is</span> <span m="334180">offered</span> <span m="334510">exactly</span> <span m="335040">once.</span> </p>
<p><span m="336700">So</span> <span m="336910">what</span> <span m="337000">does</span> <span m="337170">it</span> <span m="337290">mean?</span> <span m="338200">So</span> <span m="338390">let's</span> <span m="339320">write</span> <span m="339490">this</span> <span m="339610">down</span> <span m="339860">actually.</span> <span m="340240">So</span> <span m="341290">since</span> <span m="341870">every</span> <span m="343920">edge</span> <span m="345830">is</span> <span m="346140">traversed</span> <span m="346560">once--</span> <span m="353650">every</span> <span m="353950">edge</span> <span m="354320">in</span> <span m="356530">E</span> <span m="357720">is</span> <span m="357870">traversed</span> <span m="358510">once--</span> <span m="359250">what</span> <span m="359400">can we</span> <span m="359600">conclude</span> <span m="359970">from</span> <span m="360140">that?</span> <span m="361020">Can</span> <span m="361180">we</span> <span m="361260">say</span> <span m="361490">something</span> <span m="362010">about,</span> <span m="364000">given</span> <span m="366530">such</span> <span m="366910">as</span> <span m="368700">walk of</span> <span m="369180">length</span> <span m="369970">k,</span> <span m="371980">what</span> <span m="372130">can</span> <span m="372230">we</span> <span m="372340">say</span> <span m="372600">about</span> <span m="373150">the</span> <span m="373250">number</span> <span m="373480">of</span> <span m="373600">edges,</span> <span m="373940">for</span> <span m="374090">example?</span> </p>
<p><span m="374560">What</span> <span m="374730">can</span> <span m="374870">we</span> <span m="374960">say</span> <span m="375250">about</span> <span m="378440">the</span> <span m="378560">degree</span> <span m="379070">of</span> <span m="379630">the</span> <span m="379790">vertices?</span> <span m="380250">Because</span> <span m="380520">that's</span> <span m="380680">what</span> <span m="380820">you</span> <span m="380930">want</span> <span m="381060">to</span> <span m="381140">show,</span> <span m="381410">right?</span> <span m="381610">You</span> <span m="381710">want</span> <span m="381820">to</span> <span m="381900">show</span> <span m="382120">that</span> <span m="382200">every</span> <span m="382430">vertex</span> <span m="382840">has</span> <span m="383190">even</span> <span m="383490">degree.</span> <span m="385270">Does</span> <span m="385400">anyone</span> <span m="385720">know</span> <span m="385940">how</span> <span m="386210">to</span> <span m="387020">go</span> <span m="387200">ahead</span> <span m="387470">here?</span> <span m="387690">So</span> <span m="387860">we</span> <span m="387970">know</span> <span m="388680">that</span> <span m="388950">every</span> <span m="389310">single</span> <span m="389710">edge</span> <span m="390090">in</span> <span m="390350">E</span> <span m="391670">is</span> <span m="391890">referred</span> <span m="392330">ones</span> <span m="392670">within</span> <span m="392960">this</span> <span m="393120">whole</span> <span m="393370">walk.</span> <span m="394340">So</span> <span m="394570">what kind</span> <span m="394720">of</span> <span m="394800">properties</span> <span m="395250">can</span> <span m="395420">we derive?</span> </p>
<p><span m="401998">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="404430">PROFESSOR: That's</span> <span m="404680">true.</span> </p>
<p><span m="406667">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="411657">PROFESSOR: Maybe</span> <span m="412156">someone</span> <span m="412655">else</span> <span m="413154">can</span> <span m="413653">pick up here.</span> <span m="414152">So every</span> <span m="414651">vertex</span> <span m="417146">that</span> <span m="417645">I have</span> <span m="418144">here,</span> <span m="418643">I</span> <span m="419160">will enter</span> <span m="419640">it.</span> <span m="419700">But I also</span> <span m="420000">leave it.</span> <span m="420300">So</span> <span m="421990">if</span> <span m="422200">I</span> <span m="422290">see</span> <span m="422580">a</span> <span m="422680">vertex</span> <span m="423090">in</span> <span m="423200">here,</span> <span m="424450">I</span> <span m="424580">can</span> <span m="424760">see</span> <span m="424950">at</span> <span m="425040">least</span> <span m="425320">two</span> <span m="425870">contributing</span> <span m="426470">edges</span> <span m="427120">that</span> <span m="427350">are</span> <span m="427530">coming</span> <span m="427830">from</span> <span m="428060">this.</span> </p>
<p><span m="430330">AUDIENCE: Then</span> <span m="430690">you know</span> <span m="430820">that</span> <span m="431175">that</span> <span m="431530">note</span> <span m="431710">has</span> <span m="431950">at</span> <span m="432060">least</span> <span m="432360">degree</span> <span m="432640">2,</span> <span m="433140">but</span> <span m="433240">it</span> <span m="433320">can't</span> <span m="433570">be</span> <span m="433680">more</span> <span m="433910">than</span> <span m="434100">2</span> <span m="434440">because</span> <span m="434690">otherwise</span> <span m="435080">that</span> <span m="435330">would be</span> <span m="435585">the endpoint.</span> </p>
<p><span m="437395">PROFESSOR: Ah.</span> <span m="437730">But</span> <span m="439330">is</span> <span m="439540">it</span> <span m="439660">true</span> <span m="439910">though</span> <span m="440170">that--</span> <span m="441670">so</span> <span m="442430">for</span> <span m="442550">example,</span> <span m="443010">this</span> <span m="443060">particular</span> <span m="443460">note</span> <span m="443740">here</span> <span m="444610">may</span> <span m="444850">repeat</span> <span m="445190">itself</span> <span m="445490">somewhere</span> <span m="445800">else.</span> <span m="446120">I</span> <span m="446280">only</span> <span m="446600">have</span> <span m="446880">a</span> <span m="446950">condition</span> <span m="447900">on</span> <span m="449000">the</span> <span m="449120">Euler</span> <span m="449400">tour</span> <span m="449830">that</span> <span m="450190">every</span> <span m="450430">edge</span> <span m="450880">occurs</span> <span m="451250">exactly</span> <span m="451680">ones.</span> <span m="454340">So</span> <span m="454540">how</span> <span m="454740">could</span> <span m="454940">I</span> <span m="455270">formally</span> <span m="455720">sort</span> <span m="455900">of</span> <span m="455970">continue</span> <span m="456390">to</span> <span m="456430">prove?</span> </p>
<p><span m="457470">AUDIENCE: Well, it will</span> <span m="457830">intersect</span> <span m="458320">itself</span> <span m="458680">again,</span> <span m="459020">and</span> <span m="459895">you'll</span> <span m="460180">leave</span> <span m="460380">again</span> <span m="460700">and</span> <span m="460980">it will</span> <span m="461080">still</span> <span m="461360">have</span> <span m="461610">the even</span> <span m="462093">number.</span> </p>
<p><span m="462576">PROFESSOR: Yeah.</span> <span m="463060">That's</span> <span m="463330">true.</span> <span m="463770">That's</span> <span m="464140">right.</span> <span m="465410">Any other</span> <span m="465905">additions?</span> </p>
<p><span m="466895">AUDIENCE: You can</span> <span m="467390">define</span> <span m="467885">the number</span> <span m="468380">of edges</span> <span m="468875">that it will test</span> <span m="469370">by</span> <span m="469865">how many</span> <span m="470360">you enter it,</span> <span m="470855">so</span> <span m="471350">the number</span> <span m="471845">of edges</span> <span m="472340">that it has</span> <span m="472835">is</span> <span m="473330">twice</span> <span m="473825">the number of</span> <span m="474320">[INAUDIBLE].</span> </p>
<p><span m="476300">PROFESSOR: Yeah.</span> <span m="476795">That's</span> <span m="477290">correct.</span> <span m="477785">[INAUDIBLE]</span> </p>
<p><span m="480260">AUDIENCE: [INAUDIBLE]</span> <span m="481745">traverse</span> <span m="482240">once,</span> <span m="483230">then</span> <span m="484715">every</span> <span m="485210">time</span> <span m="485705">you enter</span> <span m="486200">a node</span> <span m="486695">you have to</span> <span m="487190">leave it.</span> <span m="487685">So</span> <span m="488180">you can say</span> <span m="488675">that you're</span> <span m="489170">never</span> <span m="489665">going to</span> <span m="491150">leave</span> <span m="494470">a</span> <span m="494910">node</span> <span m="495310">for a node</span> <span m="495610">that it</span> <span m="495910">already</span> <span m="496410">left</span> <span m="496910">for</span> <span m="497910">from there.</span> <span m="498410">And you're never going</span> <span m="498910">to</span> <span m="499410">enter</span> <span m="499910">from a</span> <span m="500410">node you</span> <span m="500910">already entered</span> <span m="501410">from</span> <span m="501910">or left to from</span> <span m="502410">there.</span> <span m="502910">So</span> <span m="503410">that's how you can say</span> <span m="503910">that you're</span> <span m="504410">only going to</span> <span m="504910">increment</span> <span m="505410">your</span> <span m="507910">degree by</span> <span m="508410">2.</span> </p>
<p><span m="508910">PROFESSOR: That's true.</span> <span m="509370">So what</span> <span m="509570">he's</span> <span m="509875">essentially</span> <span m="510270">saying</span> <span m="510580">is</span> <span m="510790">that</span> <span m="510950">every</span> <span m="511320">edge</span> <span m="511380">as</span> <span m="511660">I</span> <span m="511720">start</span> <span m="512010">to</span> <span m="512140">count</span> <span m="512419">here,</span> <span m="512659">in</span> <span m="512740">this</span> <span m="512940">particular</span> <span m="513429">tour,</span> <span m="514299">well</span> <span m="514400">they're</span> <span m="514530">all</span> <span m="514700">different.</span> <span m="515299">So</span> <span m="515400">they</span> <span m="515549">all</span> <span m="515789">counts</span> <span m="516140">towards</span> <span m="516580">a</span> <span m="516640">degree</span> <span m="517010">of</span> <span m="517140">the</span> <span m="517190">node.</span> <span m="519179">So</span> <span m="520140">everything</span> <span m="520400">that</span> <span m="520710">you have</span> <span m="521020">been</span> <span m="521130">saying</span> <span m="521830">is</span> <span m="522559">right</span> <span m="522809">on.</span> <span m="523590">What</span> <span m="523750">we</span> <span m="523850">can</span> <span m="524020">conclude</span> <span m="524440">here</span> <span m="525140">is</span> <span m="525430">that</span> <span m="525700">if</span> <span m="525890">you</span> <span m="526060">look</span> <span m="526350">at</span> <span m="526580">a</span> <span m="527790">vertex,</span> <span m="528560">say</span> <span m="529700">vertex</span> <span m="530230">U--</span> <span m="537040">for</span> <span m="537050">example,</span> <span m="537380">somewhere</span> <span m="537700">over</span> <span m="537950">here--</span> <span m="539970">well,</span> <span m="540230">it</span> <span m="540350">may</span> <span m="540520">repeat</span> <span m="540960">itself</span> <span m="541430">over</span> <span m="541770">here.</span> <span m="542240">And</span> <span m="542360">it</span> <span m="542480">has</span> <span m="542690">an</span> <span m="542800">incoming</span> <span m="543290">and</span> <span m="543450">outgoing</span> <span m="544010">edge,</span> <span m="544510">incoming</span> <span m="544990">and</span> <span m="545160">outgoing</span> <span m="545740">edge.</span> <span m="547140">And</span> <span m="547290">I</span> <span m="547310">may</span> <span m="547560">have</span> <span m="547780">it</span> <span m="548160">somewhere</span> <span m="548520">else,</span> <span m="548780">but</span> <span m="548880">say</span> <span m="549130">it's</span> <span m="549240">just at</span> <span m="549550">those</span> <span m="549810">two.</span> <span m="551040">Well,</span> <span m="551300">I</span> <span m="551400">know that</span> <span m="551750">all</span> <span m="551960">the</span> <span m="552050">edges</span> <span m="552430">are</span> <span m="552530">different.</span> </p>
<p><span m="553570">So</span> <span m="554820">I</span> <span m="554990">can</span> <span m="555180">clearly</span> <span m="555550">count this</span> <span m="555990">now.</span> <span m="556220">I</span> <span m="556290">can</span> <span m="556460">say</span> <span m="556690">that</span> <span m="556890">the</span> <span m="557040">degree</span> <span m="558440">of</span> <span m="558710">U</span> <span m="560000">must</span> <span m="560290">be</span> <span m="560540">equal</span> <span m="561840">to</span> <span m="561970">the</span> <span m="562060">number</span> <span m="562330">of</span> <span m="562450">times</span> <span m="564250">that</span> <span m="566120">U</span> <span m="566440">actually</span> <span m="567020">appears</span> <span m="570270">in</span> <span m="570400">this</span> <span m="570590">tour.</span> <span m="577400">So</span> <span m="577570">in</span> <span m="577670">the</span> <span m="577770">tour</span> <span m="578070">from</span> <span m="578260">V0</span> <span m="578990">all</span> <span m="579170">the</span> <span m="579250">way</span> <span m="579440">to</span> <span m="579620">Vk</span> <span m="580050">minus</span> <span m="580420">1.</span> <span m="581540">And</span> <span m="581710">then</span> <span m="581860">I</span> <span m="581930">have</span> <span m="582080">to</span> <span m="582180">multiply</span> <span m="582780">times</span> <span m="583210">2.</span> </p>
<p><span m="585490">And</span> <span m="585630">what I</span> <span m="585790">said,</span> <span m="586370">it's</span> <span m="586570">exactly</span> <span m="586970">what</span> <span m="587140">you</span> <span m="587290">have been</span> <span m="587520">saying,</span> <span m="587780">all</span> <span m="588080">the edges</span> <span m="588540">occur</span> <span m="588860">exactly</span> <span m="589320">once</span> <span m="590050">of</span> <span m="590240">the</span> <span m="590320">whole</span> <span m="590600">graph.</span> <span m="591640">So I</span> <span m="591890">just</span> <span m="592180">count</span> <span m="592400">the</span> <span m="592450">number</span> <span m="592640">of</span> <span m="592750">times</span> <span m="593110">I</span> <span m="593160">see</span> <span m="593470">U</span> <span m="593650">in</span> <span m="593750">here.</span> <span m="594470">I</span> <span m="594570">come</span> <span m="594810">the</span> <span m="594830">number</span> <span m="595130">of</span> <span m="595270">edges</span> <span m="595640">that</span> <span m="596160">go</span> <span m="596370">into</span> <span m="596740">it</span> <span m="596920">and</span> <span m="597130">leave</span> <span m="597310">from it.</span> <span m="597620">Well,</span> <span m="597930">that's</span> <span m="598150">twice</span> <span m="598590">times</span> <span m="599470">the</span> <span m="599580">number</span> <span m="601400">of</span> <span m="601560">times</span> <span m="601860">that</span> <span m="601960">U</span> <span m="602160">actually</span> <span m="602500">appears in this</span> <span m="602990">tour.</span> <span m="604060">So</span> <span m="604230">this</span> <span m="605530">implication</span> <span m="605875">is</span> <span m="606220">pretty</span> <span m="606450">straightforward</span> <span m="607560">because</span> <span m="607820">now</span> <span m="608010">we</span> <span m="608590">actually</span> <span m="610470">note</span> <span m="610900">that</span> <span m="611030">the</span> <span m="611120">degree</span> <span m="611410">of</span> <span m="611520">U</span> <span m="611840">is</span> <span m="612030">even.</span> <span m="612630">So</span> <span m="612770">that's</span> <span m="612980">great.</span> </p>
<p><span m="614130">So</span> <span m="614320">let's</span> <span m="614930">talk</span> <span m="615140">about</span> <span m="615310">the</span> <span m="615400">other</span> <span m="615530">implication</span> <span m="617390">and</span> <span m="617600">see</span> <span m="617770">whether we</span> <span m="617940">can</span> <span m="618440">find</span> <span m="618710">a</span> <span m="618810">trick</span> <span m="619250">to</span> <span m="619450">make</span> <span m="619640">that</span> <span m="619800">happen.</span> <span m="621490">So</span> <span m="621910">what</span> <span m="622030">do</span> <span m="622110">we</span> <span m="622230">need</span> <span m="622390">to</span> <span m="623170">start</span> <span m="623510">off</span> <span m="623730">with?</span> <span m="625150">Well,</span> <span m="626550">now</span> <span m="626750">we</span> <span m="626850">have</span> <span m="626950">to</span> <span m="627050">assume</span> <span m="627350">that</span> <span m="627590">every vertex</span> <span m="627960">has</span> <span m="628220">even</span> <span m="628530">degree.</span> <span m="629020">So</span> <span m="629230">let's</span> <span m="629420">write</span> <span m="629600">this</span> <span m="629790">down.</span> <span m="631010">So</span> <span m="631320">say</span> <span m="631570">we</span> <span m="631670">have</span> <span m="631860">the</span> <span m="631930">graph.</span> <span m="635030">And</span> <span m="635500">for</span> <span m="635720">this</span> <span m="635900">graph</span> <span m="636310">we</span> <span m="636460">actually</span> <span m="636820">issue</span> <span m="640010">that</span> <span m="642320">the</span> <span m="642410">degree</span> <span m="642970">of</span> <span m="643340">every</span> <span m="643870">vertex</span> <span m="644330">V</span> <span m="645240">is</span> <span m="645520">even.</span> </p>
<p><span m="652670">Well,</span> <span m="652980">what</span> <span m="653090">can</span> <span m="653240">we</span> <span m="653320">do?</span> <span m="658300">Well,</span> <span m="658440">this</span> <span m="658620">is</span> <span m="658680">sort</span> <span m="658880">of</span> <span m="659040">creative</span> <span m="659400">trick,</span> <span m="659720">so</span> <span m="659830">let</span> <span m="659980">me</span> <span m="660060">just</span> <span m="660490">continue</span> <span m="660785">here.</span> <span m="661970">So</span> <span m="662330">what</span> <span m="662550">we</span> <span m="662630">have</span> <span m="662980">is</span> <span m="664310">if</span> <span m="664500">you</span> <span m="664600">start</span> <span m="664830">to</span> <span m="664900">consider</span> <span m="665340">a</span> <span m="665410">walk,</span> <span m="666360">W,</span> <span m="667870">that</span> <span m="669700">touches,</span> <span m="670360">say,</span> <span m="671250">a</span> <span m="671370">lot</span> <span m="671820">of</span> <span m="671930">vertices</span> <span m="673900">in</span> <span m="674010">such</span> <span m="674350">a</span> <span m="674450">way</span> <span m="675830">that</span> <span m="676710">W</span> <span m="677450">is</span> <span m="677610">actually</span> <span m="677870">the</span> <span m="678030">longest</span> <span m="678560">walk.</span> <span m="680500">So</span> <span m="680630">that's</span> <span m="680840">often</span> <span m="681420">what</span> <span m="681930">you</span> <span m="682030">do</span> <span m="682100">in</span> <span m="682160">graph</span> <span m="682480">theory.</span> <span m="683130">You</span> <span m="683450">think</span> <span m="683690">about,</span> <span m="684000">say,</span> <span m="684160">the</span> <span m="684260">shortest</span> <span m="685600">path</span> <span m="686040">or</span> <span m="686200">whatever</span> <span m="686350">with</span> <span m="686520">a</span> <span m="686670">certain</span> <span m="686840">property,</span> <span m="687300">or</span> <span m="687350">the</span> <span m="687450">longest</span> <span m="687910">walk,</span> <span m="688240">or</span> <span m="688360">whatever</span> <span m="689130">with a</span> <span m="689210">certain</span> <span m="689500">property.</span> <span m="689960">So that's</span> <span m="690210">what</span> <span m="690330">you're</span> <span m="690460">doing</span> <span m="690730">here.</span> </p>
<p><span m="691760">So</span> <span m="691830">let</span> <span m="691960">W</span> <span m="692340">be</span> <span m="692490">the</span> <span m="692730">longest</span> <span m="693240">walk.</span> <span m="698020">And</span> <span m="698780">well,</span> <span m="698960">we</span> <span m="699090">want</span> <span m="699280">to</span> <span m="699380">prove</span> <span m="699710">something</span> <span m="700940">about</span> <span m="701450">an</span> <span m="701890">Euler</span> <span m="702080">tour.</span> <span m="702420">An</span> <span m="702560">Euler</span> <span m="702760">tour</span> <span m="702850">has</span> <span m="703330">a very</span> <span m="703480">specific</span> <span m="703910">property</span> <span m="704480">that all</span> <span m="704670">the</span> <span m="704810">edges</span> <span m="705120">occur</span> <span m="705470">only</span> <span m="705770">once.</span> <span m="706420">So</span> <span m="706550">it</span> <span m="706650">makes</span> <span m="706950">sense</span> <span m="707260">to</span> <span m="707390">look</span> <span m="707590">at</span> <span m="707670">walks</span> <span m="711220">in</span> <span m="711370">which</span> <span m="711690">all</span> <span m="711750">the</span> <span m="711950">edges</span> <span m="713840">of</span> <span m="714080">which</span> <span m="714310">it</span> <span m="714390">walks</span> <span m="715700">are</span> <span m="715930">unique,</span> <span m="716280">are</span> <span m="716930">occurring</span> <span m="717300">only</span> <span m="717530">once.</span> <span m="717980">So</span> <span m="719470">we're</span> <span m="719620">interested in</span> <span m="720090">the</span> <span m="720180">longest</span> <span m="720600">walk</span> <span m="721540">that</span> <span m="722490">traverses</span> <span m="727950">no</span> <span m="728220">edge</span> <span m="731400">more</span> <span m="732970">than</span> <span m="733220">once.</span> <span m="735930">So</span> <span m="736120">that's</span> <span m="738040">important.</span> </p>
<p><span m="742060">Now,</span> <span m="742200">let's</span> <span m="743320">think</span> <span m="743580">about</span> <span m="743820">this</span> <span m="744010">a</span> <span m="744040">little</span> <span m="744270">bit.</span> <span m="745510">So</span> <span m="745600">the</span> <span m="745700">first</span> <span m="746080">property</span> <span m="746350">that you</span> <span m="746620">may</span> <span m="746800">want</span> <span m="747020">to</span> <span m="747460">consider</span> <span m="748020">it--</span> <span m="749070">you</span> <span m="749230">really</span> <span m="749450">want</span> <span m="749590">to</span> <span m="749670">show</span> <span m="749910">that</span> <span m="750000">W</span> <span m="750280">is</span> <span m="750560">going</span> <span m="750740">to</span> <span m="750830">be</span> <span m="750940">an</span> <span m="751060">Euler</span> <span m="751460">tour.</span> <span m="752230">That's</span> <span m="752450">what we're</span> <span m="752660">going</span> <span m="752900">to</span> <span m="753010">try</span> <span m="753250">to</span> <span m="753350">prove</span> <span m="753700">here.</span> <span m="754880">So</span> <span m="754920">the</span> <span m="755010">first</span> <span m="755290">thing</span> <span m="755730">we may</span> <span m="755910">want</span> <span m="756080">to</span> <span m="756170">show</span> <span m="756560">is</span> <span m="756700">that</span> <span m="756850">actually</span> <span m="758290">it</span> <span m="758460">goes</span> <span m="758720">all</span> <span m="758910">around.</span> <span m="759270">So</span> <span m="759800">V0</span> <span m="760470">is</span> <span m="760600">equal</span> <span m="760850">to</span> <span m="761000">Vk.</span> <span m="762090">That</span> <span m="762210">will</span> <span m="762300">be</span> <span m="762400">great</span> <span m="762770">if</span> <span m="762880">we</span> <span m="762960">can</span> <span m="763100">do</span> <span m="763240">this.</span> </p>
<p><span m="764540">So</span> <span m="764690">we</span> <span m="764800">want</span> <span m="764920">to</span> <span m="765000">show</span> <span m="765380">that</span> <span m="765872">this</span> <span m="767350">is</span> <span m="767500">true.</span> <span m="768720">So</span> <span m="770650">what</span> <span m="770840">would</span> <span m="770980">happen</span> <span m="771370">if</span> <span m="771500">this</span> <span m="771710">would</span> <span m="771880">not</span> <span m="772110">be</span> <span m="772230">the</span> <span m="772380">case?</span> <span m="775580">So</span> <span m="775800">suppose</span> <span m="778290">these</span> <span m="778570">two</span> <span m="778760">would</span> <span m="778950">not</span> <span m="779120">be</span> <span m="779310">equal</span> <span m="779510">to</span> <span m="779640">one</span> <span m="779900">another.</span> <span m="785860">Well,</span> <span m="788690">actually,</span> <span m="789160">I'm</span> <span m="789600">skipping</span> <span m="790000">a</span> <span m="790050">little</span> <span m="790210">bit</span> <span m="790380">ahead</span> <span m="790690">here</span> <span m="791010">in the</span> <span m="791150">proof,</span> <span m="792150">I</span> <span m="792270">notice.</span> <span m="794570">So</span> <span m="794790">this</span> <span m="795020">will</span> <span m="795210">be</span> <span m="795560">the</span> <span m="795650">conclusion</span> <span m="796240">of</span> <span m="796360">another</span> <span m="796600">observation.</span> <span m="797380">So</span> <span m="797730">let me</span> <span m="797830">start</span> <span m="798270">with</span> <span m="799670">the</span> <span m="799770">first</span> <span m="800170">observation.</span> </p>
<p><span m="802940">So</span> <span m="803150">first</span> <span m="803450">of</span> <span m="803550">all,</span> <span m="805370">let</span> <span m="805490">us</span> <span m="806560">consider</span> <span m="806890">this</span> <span m="807080">end</span> <span m="807300">node</span> <span m="807570">over</span> <span m="807820">here,</span> <span m="807960">this</span> <span m="808020">end</span> <span m="808200">vertex,</span> <span m="808460">Vk.</span> <span m="810300">Now</span> <span m="811090">if</span> <span m="811960">there</span> <span m="812760">is</span> <span m="813090">an</span> <span m="813300">edge</span> <span m="814370">that</span> <span m="815290">is</span> <span m="815520">not</span> <span m="815820">covered</span> <span m="816400">in</span> <span m="816590">this</span> <span m="816820">walk--</span> <span m="817800">and</span> <span m="817950">I</span> <span m="818000">can</span> <span m="818170">lengthen</span> <span m="818560">the</span> <span m="818660">walk,</span> <span m="818890">right?</span> <span m="819860">So</span> <span m="820210">if</span> <span m="820430">I</span> <span m="820530">have,</span> <span m="822000">say,</span> <span m="822290">another</span> <span m="822610">edge,</span> <span m="823440">which</span> <span m="824080">looks</span> <span m="824530">like</span> <span m="824830">Vk,</span> <span m="827920">an</span> <span m="828110">edge</span> <span m="828260">from</span> <span m="828470">Vk</span> <span m="828910">to</span> <span m="829020">some</span> <span m="829260">U,</span> <span m="830170">where</span> <span m="831730">this</span> <span m="832210">edge</span> <span m="832770">is</span> <span m="833750">not</span> <span m="835680">in</span> <span m="836440">the</span> <span m="836570">walk.</span> <span m="839600">Or</span> <span m="841070">the</span> <span m="841180">other</span> <span m="841370">one</span> <span m="841560">is not</span> <span m="842000">in the</span> <span m="842300">walk.</span> <span m="842820">Well,</span> <span m="843030">it doesn't</span> <span m="843190">matter</span> <span m="843540">in</span> <span m="844200">an</span> <span m="844270">indirect</span> <span m="844570">graph,</span> <span m="844840">of</span> <span m="844960">course.</span> </p>
<p><span m="846450">So</span> <span m="846610">if</span> <span m="846770">this</span> <span m="846960">is</span> <span m="847090">not</span> <span m="847270">in</span> <span m="847370">the</span> <span m="847460">walk,</span> <span m="848010">then</span> <span m="848140">what</span> <span m="848260">can</span> <span m="848400">I</span> <span m="848480">do?</span> <span m="848630">I can</span> <span m="848940">just</span> <span m="849170">lengthen</span> <span m="849550">it,</span> <span m="849845">right?</span> <span m="850140">I</span> <span m="850270">can</span> <span m="850450">just</span> <span m="850660">say</span> <span m="850890">V0</span> <span m="851610">goes</span> <span m="851880">to</span> <span m="851980">V1</span> <span m="853270">goes</span> <span m="853500">to</span> <span m="853610">Vk</span> <span m="854305">goes</span> <span m="855190">to</span> <span m="855400">U.</span> <span m="856130">Now</span> <span m="856300">I</span> <span m="856370">have</span> <span m="856570">a</span> <span m="856640">longer</span> <span m="857090">walk.</span> <span m="859590">And</span> <span m="860170">that's</span> <span m="860390">a</span> <span m="860440">contradiction.</span> <span m="862340">So</span> <span m="862820">I</span> <span m="862960">know</span> <span m="864040">that</span> <span m="865860">all</span> <span m="866120">the</span> <span m="866290">edges</span> <span m="869332">in</span> <span m="869750">which</span> <span m="870100">Vk</span> <span m="870555">is</span> <span m="870820">participating</span> <span m="871660">are</span> <span m="871840">actually</span> <span m="872310">covered</span> <span m="873000">by</span> <span m="873210">the</span> <span m="873330">walk.</span> <span m="873910">So</span> <span m="873990">that's</span> <span m="874150">a</span> <span m="874200">great</span> <span m="874780">property.</span> <span m="875360">So</span> <span m="876090">let's</span> <span m="876290">write</span> <span m="876480">this</span> <span m="876660">down.</span> <span m="876930">So</span> <span m="878530">all</span> <span m="879000">edges</span> <span m="885330">that</span> <span m="885500">are</span> <span m="885790">incident</span> <span m="889390">to</span> <span m="890370">Vk</span> <span m="892150">are</span> <span m="892430">actually</span> <span m="892860">used--</span> <span m="895880">traversed--</span> <span m="897350">in</span> <span m="898170">the</span> <span m="898300">walk,</span> <span m="898590">W.</span> </p>
<p><span m="900370">So</span> <span m="900490">now</span> <span m="900660">let's</span> <span m="900900">have</span> <span m="901080">a</span> <span m="901180">look</span> <span m="902420">whether</span> <span m="902780">we</span> <span m="902950">can</span> <span m="903180">show</span> <span m="904120">that</span> <span m="905730">it's</span> <span m="906960">a</span> <span m="907070">walk</span> <span m="907420">that goes</span> <span m="907650">all</span> <span m="907960">the</span> <span m="908060">way</span> <span m="908130">around.</span> <span m="908710">So</span> <span m="908900">let's</span> <span m="909990">try</span> <span m="910240">to</span> <span m="910360">prove</span> <span m="910740">this</span> <span m="910900">statement.</span> <span m="912460">So</span> <span m="912680">let</span> <span m="912840">me</span> <span m="913010">repeat</span> <span m="913390">this</span> <span m="914440">over</span> <span m="914790">here.</span> <span m="917060">So</span> <span m="917180">what</span> <span m="917340">we</span> <span m="917410">want</span> <span m="917570">to</span> <span m="917640">show</span> <span m="918020">is</span> <span m="918660">that</span> <span m="919020">Vk</span> <span m="919930">equals</span> <span m="920550">V0.</span> <span m="922200">So</span> <span m="922400">given</span> <span m="922710">this</span> <span m="922860">statement,</span> <span m="923770">do</span> <span m="924370">you</span> <span m="924500">have</span> <span m="924610">any</span> <span m="924900">idea</span> <span m="925260">how</span> <span m="925440">we</span> <span m="925580">could</span> <span m="925830">possibly</span> <span m="926320">prove</span> <span m="926670">this,</span> <span m="927210">also</span> <span m="927510">given</span> <span m="927840">what</span> <span m="928010">we</span> <span m="928140">have</span> <span m="928430">been</span> <span m="928600">doing</span> <span m="928850">over</span> <span m="929120">here?</span> </p>
<p><span m="931170">Suppose</span> <span m="931650">this</span> <span m="931800">would</span> <span m="931990">not</span> <span m="932210">be</span> <span m="932340">the</span> <span m="932470">case,</span> <span m="932820">right?</span> <span m="935910">So</span> <span m="936510">suppose</span> <span m="939870">the</span> <span m="940030">start</span> <span m="940640">and</span> <span m="940770">the end</span> <span m="941100">node</span> <span m="941390">in</span> <span m="941550">this</span> <span m="941870">walk</span> <span m="942240">are</span> <span m="942350">not</span> <span m="942620">equal</span> <span m="942910">to</span> <span m="943010">one</span> <span m="943320">another.</span> <span m="945970">So</span> <span m="947140">what</span> <span m="947280">can</span> <span m="947420">we</span> <span m="947520">do</span> <span m="947710">here?</span> <span m="950180">So</span> <span m="950230">maybe</span> <span m="950430">there's</span> <span m="950640">some</span> <span m="952120">suggestions?</span> <span m="953500">I</span> <span m="953895">already</span> <span m="954290">had you</span> <span m="954500">once.</span> <span m="956560">Maybe</span> <span m="956740">someone</span> <span m="957040">else?</span> </p>
<p><span m="961624">AUDIENCE: It</span> <span m="962560">has</span> <span m="962850">to</span> <span m="963190">attach</span> <span m="963600">to</span> <span m="963710">another</span> <span m="964410">vertex.</span> <span m="965060">And</span> <span m="965220">you</span> <span m="965300">know</span> <span m="965460">that</span> <span m="965610">all</span> <span m="965770">the</span> <span m="965870">degrees</span> <span m="966520">are</span> <span m="966650">even.</span> <span m="967460">So</span> <span m="968375">it means</span> <span m="968660">that</span> <span m="969320">Vk</span> <span m="969750">must</span> <span m="969960">attach</span> <span m="970290">to</span> <span m="970860">V0.</span> </p>
<p><span m="973750">PROFESSOR: So</span> <span m="973910">what</span> <span m="974040">you're</span> <span m="974160">saying,</span> <span m="974530">first</span> <span m="974800">of</span> <span m="974880">all,</span> <span m="975280">is that</span> <span m="976800">if</span> <span m="976970">this</span> <span m="977160">would</span> <span m="977310">not</span> <span m="977500">be</span> <span m="977590">the</span> <span m="977730">case,</span> <span m="979270">then</span> <span m="979660">you</span> <span m="983290">know</span> <span m="983590">that</span> <span m="983680">the</span> <span m="983770">degrees</span> <span m="984220">must be</span> <span m="984630">even.</span> <span m="986070">But</span> <span m="986180">suppose</span> <span m="986510">this</span> <span m="986660">would</span> <span m="986750">not</span> <span m="986900">be</span> <span m="987010">the</span> <span m="987130">case.</span> <span m="987630">That</span> <span m="987780">would</span> <span m="987910">mean</span> <span m="988140">actually</span> <span m="988490">that</span> <span m="988670">the</span> <span m="988710">degree</span> <span m="989200">would</span> <span m="989360">be</span> <span m="989500">odd,</span> <span m="989805">right?</span> <span m="992310">OK.</span> <span m="992590">So</span> <span m="992750">let's</span> <span m="992950">think</span> <span m="993140">about</span> <span m="993370">that</span> <span m="993470">a</span> <span m="993510">little</span> <span m="993750">bit.</span> <span m="994600">So</span> <span m="994730">let's</span> <span m="995000">write</span> <span m="995140">this</span> <span m="995340">down.</span> </p>
<p><span m="996380">So</span> <span m="996600">otherwise</span> <span m="997410">the</span> <span m="997500">Vk</span> <span m="998700">has</span> <span m="999750">odd</span> <span m="1000100">degree.</span> <span m="1002010">But</span> <span m="1003460">the</span> <span m="1003600">only</span> <span m="1003880">node</span> <span m="1004430">has</span> <span m="1004750">odd</span> <span m="1004890">degree</span> <span m="1005280">in</span> <span m="1005410">this</span> <span m="1005620">particular</span> <span m="1006060">walk,</span> <span m="1006330">W.</span> <span m="1008270">So</span> <span m="1008470">that's</span> <span m="1008670">what</span> <span m="1008820">we</span> <span m="1008940">do</span> <span m="1009130">now.</span> <span m="1010140">Because</span> <span m="1012670">we</span> <span m="1012830">really</span> <span m="1013100">consider</span> <span m="1014840">this</span> <span m="1016020">walk.</span> <span m="1021490">So</span> <span m="1021790">we</span> <span m="1021920">see</span> <span m="1022250">that</span> <span m="1023110">if</span> <span m="1023360">I</span> <span m="1023810">have</span> <span m="1024119">V0</span> <span m="1025150">unequal</span> <span m="1025380">to</span> <span m="1025560">Vk</span> <span m="1026490">and</span> <span m="1026720">Vk</span> <span m="1027260">maybe</span> <span m="1027560">repeated</span> <span m="1028079">in</span> <span m="1028250">this</span> <span m="1028440">walk</span> <span m="1028839">a</span> <span m="1028920">number</span> <span m="1029230">of</span> <span m="1029369">times,</span> <span m="1030240">each</span> <span m="1030540">time</span> <span m="1030780">it</span> <span m="1030900">is</span> <span m="1031030">repeated</span> <span m="1031470">it has</span> <span m="1031730">an</span> <span m="1031859">incoming</span> <span m="1032400">and an</span> <span m="1032609">outgoing</span> <span m="1033250">edge.</span> <span m="1034390">And</span> <span m="1034770">all</span> <span m="1035040">these</span> <span m="1035290">edges</span> <span m="1035579">are</span> <span m="1035980">different</span> <span m="1036520">because</span> <span m="1038089">they</span> <span m="1038220">do</span> <span m="1038329">not</span> <span m="1038510">occur</span> <span m="1038800">more</span> <span m="1039000">than</span> <span m="1039140">once.</span> <span m="1039790">So</span> <span m="1040319">whenever</span> <span m="1040500">Vk</span> <span m="1041500">enters</span> <span m="1041980">in</span> <span m="1042119">this</span> <span m="1042270">middle</span> <span m="1042619">part</span> <span m="1042980">over</span> <span m="1043240">here--</span> <span m="1044680">it's</span> <span m="1045250">partaking</span> <span m="1045780">over</span> <span m="1046020">here--</span> <span m="1046810">it</span> <span m="1047230">gives</span> <span m="1047440">an</span> <span m="1047609">even</span> <span m="1047930">contribution</span> <span m="1048520">to</span> <span m="1048620">the</span> <span m="1048720">number</span> <span m="1048960">of</span> <span m="1049100">edges</span> <span m="1049520">in</span> <span m="1049670">this</span> <span m="1049840">walk,</span> <span m="1050830">and</span> <span m="1051740">it</span> <span m="1051900">has</span> <span m="1052150">one</span> <span m="1052370">extra</span> <span m="1052770">incoming</span> <span m="1053280">edge</span> <span m="1053540">over</span> <span m="1053810">here.</span> <span m="1054050">So</span> <span m="1054270">if</span> <span m="1054600">V0</span> <span m="1055900">is</span> <span m="1056120">not</span> <span m="1056310">equal</span> <span m="1056590">to</span> <span m="1056760">Vk,</span> <span m="1058390">we</span> <span m="1058550">will</span> <span m="1058750">have</span> <span m="1059220">an</span> <span m="1059440">odd</span> <span m="1059600">degree</span> <span m="1060650">of</span> <span m="1063050">edges</span> <span m="1064300">in</span> <span m="1064490">W,</span> <span m="1065500">in</span> <span m="1065710">which</span> <span m="1066270">Vk</span> <span m="1066880">is</span> <span m="1067250">participating.</span> </p>
<p><span m="1069010">So</span> <span m="1069070">now</span> <span m="1069260">we</span> <span m="1069400">can</span> <span m="1069940">go</span> <span m="1070140">ahead</span> <span m="1070540">and we</span> <span m="1070650">can</span> <span m="1070940">have</span> <span m="1071130">a</span> <span m="1071220">look</span> <span m="1071430">over</span> <span m="1071690">here</span> <span m="1072500">because</span> <span m="1073660">we</span> <span m="1073780">just</span> <span m="1074110">showed</span> <span m="1074390">here that</span> <span m="1074730">all</span> <span m="1075000">the</span> <span m="1075150">edges</span> <span m="1075740">incident</span> <span m="1076220">to</span> <span m="1076370">Vk</span> <span m="1077100">are</span> <span m="1077330">actually</span> <span m="1077670">used in</span> <span m="1078040">W.</span> <span m="1078960">So</span> <span m="1079230">we</span> <span m="1079330">can now</span> <span m="1079660">conclude</span> <span m="1080720">that</span> <span m="1081570">this</span> <span m="1081790">means</span> <span m="1081990">that</span> <span m="1082330">Vk</span> <span m="1082670">also</span> <span m="1083080">has</span> <span m="1083310">odd</span> <span m="1083430">degree</span> <span m="1083780">in</span> <span m="1084070">G</span> <span m="1084920">because</span> <span m="1085260">all</span> <span m="1085500">the</span> <span m="1085780">edges in</span> <span m="1086200">G,</span> <span m="1087550">in</span> <span m="1087690">which</span> <span m="1087930">Vk</span> <span m="1088270">is</span> <span m="1088510">participating,</span> <span m="1089540">are</span> <span m="1089890">actually</span> <span m="1090280">used</span> <span m="1090560">in</span> <span m="1090720">the</span> <span m="1090820">walk.</span> <span m="1091810">So I hope</span> <span m="1092200">that is</span> <span m="1092600">clear.</span> </p>
<p><span m="1095670">So</span> <span m="1095800">let's</span> <span m="1096030">write</span> <span m="1096200">this</span> <span m="1096370">down.</span> <span m="1098840">So</span> <span m="1099430">we'll</span> <span m="1099640">use</span> <span m="1101310">the</span> <span m="1101410">first</span> <span m="1101780">statement</span> <span m="1102250">over</span> <span m="1102370">here, so</span> <span m="1102770">by</span> <span m="1102990">1.</span> <span m="1104070">We</span> <span m="1104280">have</span> <span m="1104610">that</span> <span m="1104750">Vk</span> <span m="1105810">has</span> <span m="1106940">odd</span> <span m="1109140">degree</span> <span m="1111190">in</span> <span m="1111600">G.</span> <span m="1112410">And</span> <span m="1112480">now</span> <span m="1112650">we</span> <span m="1112790">come</span> <span m="1113020">to</span> <span m="1113150">what you</span> <span m="1113360">were</span> <span m="1113540">saying.</span> <span m="1114160">We</span> <span m="1114340">have</span> <span m="1114500">assumed</span> <span m="1115010">over</span> <span m="1115320">here</span> <span m="1116120">that</span> <span m="1116580">the</span> <span m="1116690">degree</span> <span m="1117150">is</span> <span m="1117370">even.</span> <span m="1118490">So</span> <span m="1118910">this</span> <span m="1119090">cannot</span> <span m="1119340">be</span> <span m="1119440">the</span> <span m="1119550">case,</span> <span m="1120190">so</span> <span m="1120360">that's</span> <span m="1120670">a</span> <span m="1121300">contradiction.</span> <span m="1122000">So</span> <span m="1122220">we</span> <span m="1122320">know</span> <span m="1122950">that</span> <span m="1123760">the</span> <span m="1123950">otherwise</span> <span m="1124430">situation</span> <span m="1125070">cannot</span> <span m="1125370">occur.</span> <span m="1125740">So</span> <span m="1125920">we</span> <span m="1126340">know</span> <span m="1126640">that</span> <span m="1126915">Vk</span> <span m="1127190">must</span> <span m="1127440">be</span> <span m="1127600">equal</span> <span m="1127790">to</span> <span m="1127900">V0.</span> <span m="1128960">So</span> <span m="1129160">that's</span> <span m="1130500">proving</span> <span m="1130960">that we</span> <span m="1131140">actually</span> <span m="1131540">have</span> <span m="1131690">a</span> <span m="1131790">walk</span> <span m="1132030">that</span> <span m="1132120">goes</span> <span m="1132910">all</span> <span m="1133170">around.</span> </p>
<p><span m="1134380">Now,</span> <span m="1134520">we</span> <span m="1134640">are</span> <span m="1134690">not</span> <span m="1135160">yet</span> <span m="1135460">done.</span> <span m="1136710">Because</span> <span m="1137050">why</span> <span m="1137140">would</span> <span m="1137740">this</span> <span m="1137980">be</span> <span m="1139560">an</span> <span m="1139990">Euler</span> <span m="1140440">tour?</span> <span m="1140790">For</span> <span m="1140960">an</span> <span m="1141300">Euler tour,</span> <span m="1141780">we</span> <span m="1141890">really</span> <span m="1142160">want</span> <span m="1142465">that</span> <span m="1142770">every</span> <span m="1143160">single</span> <span m="1143610">edge</span> <span m="1146806">that</span> <span m="1147550">G</span> <span m="1147800">has</span> <span m="1148390">is</span> <span m="1148580">actually</span> <span m="1148980">used</span> <span m="1149320">in</span> <span m="1149450">the</span> <span m="1149540">tour.</span> <span m="1151450">So</span> <span m="1151640">we</span> <span m="1151720">need</span> <span m="1151900">much</span> <span m="1152100">more.</span> <span m="1154850">So</span> <span m="1155090">let's</span> <span m="1157460">use</span> <span m="1157610">this</span> <span m="1159112">board.</span> </p>
<p><span m="1175420">So</span> <span m="1175630">suppose</span> <span m="1178000">that</span> <span m="1178380">W</span> <span m="1179800">is</span> <span m="1180110">not</span> <span m="1181450">an</span> <span m="1181690">Euler</span> <span m="1181950">tour.</span> <span m="1182460">We're</span> <span m="1182610">going to</span> <span m="1182885">use</span> <span m="1183160">contradiction.</span> <span m="1185720">But</span> <span m="1185890">we</span> <span m="1186010">already</span> <span m="1186380">have</span> <span m="1186540">shown</span> <span m="1187620">this</span> <span m="1187780">particular</span> <span m="1188120">property,</span> <span m="1188630">so</span> <span m="1188760">we</span> <span m="1188850">can</span> <span m="1189010">use</span> <span m="1189260">this.</span> <span m="1190060">So</span> <span m="1190230">suppose</span> <span m="1190550">W</span> <span m="1190870">is</span> <span m="1191030">not</span> <span m="1191300">an</span> <span m="1191773">Euler</span> <span m="1192246">tour.</span> <span m="1192720">Well, we</span> <span m="1192830">know</span> <span m="1193880">that</span> <span m="1193990">G is</span> <span m="1194480">connected.</span> <span m="1196710">So</span> <span m="1196900">that's</span> <span m="1197140">a</span> <span m="1197180">good</span> <span m="1197360">thing</span> <span m="1197710">because</span> <span m="1199810">that</span> <span m="1200020">means</span> <span m="1201270">that</span> <span m="1201470">if</span> <span m="1201650">it is</span> <span m="1201700">not an</span> <span m="1202080">Euler</span> <span m="1202510">tour,</span> <span m="1203740">there</span> <span m="1203910">is</span> <span m="1204050">an</span> <span m="1204230">edge</span> <span m="1206030">that</span> <span m="1206200">is</span> <span m="1206480">not</span> <span m="1207210">used</span> <span m="1209510">in</span> <span m="1210250">W,</span> <span m="1210620">in</span> <span m="1210730">the</span> <span m="1210840">walk,</span> <span m="1212050">but</span> <span m="1214240">it</span> <span m="1214560">is</span> <span m="1215210">incident</span> <span m="1217765">to</span> <span m="1218200">some</span> <span m="1218710">vertex</span> <span m="1224250">that</span> <span m="1224590">is</span> <span m="1224750">used</span> <span m="1226230">in</span> <span m="1226410">W.</span> </p>
<p><span m="1228730">So</span> <span m="1228910">this</span> <span m="1229080">is</span> <span m="1229180">where</span> <span m="1229360">the</span> <span m="1229450">connectivity</span> <span m="1230340">here</span> <span m="1230510">comes in</span> <span m="1230880">play.</span> <span m="1231800">Now</span> <span m="1233410">let us</span> <span m="1233750">call</span> <span m="1233950">this</span> <span m="1234190">edge--</span> <span m="1236040">so</span> <span m="1236280">let</span> <span m="1237580">U-Vi</span> <span m="1238930">be</span> <span m="1239410">this</span> <span m="1239610">particular</span> <span m="1240100">edge.</span> <span m="1243710">So</span> <span m="1244580">U</span> <span m="1245980">is</span> <span m="1246200">not</span> <span m="1246390">used</span> <span m="1246830">in</span> <span m="1246960">the</span> <span m="1247060">walk--</span> <span m="1247850">well,</span> <span m="1248100">it's maybe</span> <span m="1248360">used</span> <span m="1248620">in</span> <span m="1248670">the</span> <span m="1248770">walk,</span> <span m="1249080">but</span> <span m="1250310">the</span> <span m="1250450">edge</span> <span m="1250630">is not</span> <span m="1250750">used</span> <span m="1251040">in</span> <span m="1251130">the</span> <span m="1251220">walk.</span> <span m="1252130">And</span> <span m="1252370">Vi</span> <span m="1253040">is</span> <span m="1253260">part</span> <span m="1253510">of</span> <span m="1253630">the</span> <span m="1253700">walk.</span> </p>
<p><span m="1254350">So</span> <span m="1255260">what</span> <span m="1255390">can</span> <span m="1255500">we</span> <span m="1255620">do</span> <span m="1255820">here?</span> <span m="1257490">So</span> <span m="1257930">does</span> <span m="1258110">anybody</span> <span m="1258480">see</span> <span m="1258720">what</span> <span m="1258900">we</span> <span m="1258990">could</span> <span m="1259170">do</span> <span m="1259350">here?</span> <span m="1260030">Can</span> <span m="1260210">we</span> <span m="1260330">somehow</span> <span m="1262030">get</span> <span m="1262200">a</span> <span m="1262260">contradiction?</span> <span m="1263140">So</span> <span m="1263690">when</span> <span m="1263850">we</span> <span m="1264200">started</span> <span m="1265360">out</span> <span m="1265580">here,</span> <span m="1266210">we</span> <span m="1266350">actually</span> <span m="1266760">assumed</span> <span m="1267260">that</span> <span m="1267623">W</span> <span m="1267986">is</span> <span m="1268350">the</span> <span m="1268440">longest</span> <span m="1268910">walk</span> <span m="1271250">in</span> <span m="1271420">which</span> <span m="1271730">the</span> <span m="1271870">edges do</span> <span m="1272300">not</span> <span m="1272480">occur</span> <span m="1273310">more</span> <span m="1273600">than</span> <span m="1273720">once.</span> <span m="1276270">Can</span> <span m="1276410">we</span> <span m="1276500">create</span> <span m="1276800">a</span> <span m="1276850">longer</span> <span m="1277140">walk</span> <span m="1277700">by</span> <span m="1277830">simply</span> <span m="1278250">using</span> <span m="1278600">this?</span> <span m="1279660">Now,</span> <span m="1279770">notice</span> <span m="1280390">that</span> <span m="1280490">we</span> <span m="1280760">did</span> <span m="1281250">prove</span> <span m="1281660">that</span> <span m="1282080">V0</span> <span m="1283310">goes</span> <span m="1284390">all</span> <span m="1284720">around</span> <span m="1286606">to</span> <span m="1287355">Vk,</span> <span m="1289360">and</span> <span m="1289550">so</span> <span m="1289750">on.</span> <span m="1290410">So</span> <span m="1291360">can</span> <span m="1291560">we</span> <span m="1291710">find</span> <span m="1292190">a</span> <span m="1292550">longer</span> <span m="1292930">walk</span> <span m="1293510">that uses</span> <span m="1293870">this</span> <span m="1294060">extra</span> <span m="1294390">edge?</span> <span m="1298020">Any</span> <span m="1299130">ideas?</span> </p>
<p><span m="1300447">What would</span> <span m="1300924">happen</span> <span m="1301401">if</span> <span m="1301880">we,</span> <span m="1302240">for</span> <span m="1302320">example,</span> <span m="1302940">started</span> <span m="1303355">a</span> <span m="1303980">walk with</span> <span m="1304280">U?</span> <span m="1305630">Let's see how</span> <span m="1305910">that would</span> <span m="1306230">work</span> <span m="1306510">out.</span> <span m="1307610">We</span> <span m="1307740">can</span> <span m="1307930">have</span> <span m="1308110">U.</span> <span m="1309310">We</span> <span m="1309480">walk</span> <span m="1309890">to</span> <span m="1310560">Vi.</span> <span m="1311570">Well,</span> <span m="1311890">let's</span> <span m="1312610">just</span> <span m="1312850">simply</span> <span m="1313280">start</span> <span m="1313820">walking</span> <span m="1314900">all</span> <span m="1315150">around.</span> <span m="1315720">So</span> <span m="1315860">essentially</span> <span m="1316340">we have</span> <span m="1316830">Vi</span> <span m="1317070">over</span> <span m="1317410">here</span> <span m="1318620">and</span> <span m="1319920">U</span> <span m="1320460">over</span> <span m="1320600">here.</span> <span m="1323630">So</span> <span m="1323970">we</span> <span m="1324050">could</span> <span m="1324210">do</span> <span m="1324460">is</span> <span m="1324760">we</span> <span m="1324860">can</span> <span m="1325540">start</span> <span m="1325920">walking</span> <span m="1326310">like</span> <span m="1326530">this</span> <span m="1328620">and</span> <span m="1328810">then</span> <span m="1329000">end</span> <span m="1329270">over</span> <span m="1329580">here.</span> <span m="1330290">And</span> <span m="1330420">notice</span> <span m="1330690">we</span> <span m="1331060">have</span> <span m="1331300">used</span> <span m="1331590">one</span> <span m="1331780">extra</span> <span m="1332160">edge.</span> </p>
<p><span m="1333310">So</span> <span m="1333480">we</span> <span m="1333580">get</span> <span m="1333730">a</span> <span m="1333790">longer</span> <span m="1334100">walk.</span> <span m="1335070">So</span> <span m="1335270">we</span> <span m="1335380">go</span> <span m="1336190">to</span> <span m="1336600">Vi</span> <span m="1337005">plus</span> <span m="1337410">1,</span> <span m="1337990">and</span> <span m="1338130">we</span> <span m="1338220">go</span> <span m="1338480">all</span> <span m="1338630">the</span> <span m="1338730">way</span> <span m="1338940">up</span> <span m="1339070">to</span> <span m="1340560">Vk,</span> <span m="1341530">which</span> <span m="1341540">is</span> <span m="1341670">equal</span> <span m="1341930">to</span> <span m="1342040">V0.</span> <span m="1343230">And</span> <span m="1343360">then</span> <span m="1343480">we</span> <span m="1343560">go</span> <span m="1343840">up</span> <span m="1343980">to</span> <span m="1344390">V1,</span> <span m="1344860">all</span> <span m="1345050">the</span> <span m="1345140">way</span> <span m="1345380">up</span> <span m="1345600">to</span> <span m="1346190">Vi.</span> <span m="1347870">This</span> <span m="1348050">is</span> <span m="1348190">a</span> <span m="1348240">longer</span> <span m="1348580">walk.</span> <span m="1349960">And</span> <span m="1350650">therefore</span> <span m="1351020">we</span> <span m="1351140">have</span> <span m="1351780">a</span> <span m="1351900">contradiction.</span> <span m="1354240">And</span> <span m="1354390">now</span> <span m="1354510">the</span> <span m="1354620">proof</span> <span m="1354850">is</span> <span m="1354980">finished</span> <span m="1355790">because</span> <span m="1356855">that</span> <span m="1357290">means</span> <span m="1357600">that</span> <span m="1357800">this</span> <span m="1358360">assumption</span> <span m="1358900">was</span> <span m="1359070">wrong.</span> <span m="1359740">So</span> <span m="1359920">W</span> <span m="1360320">actually</span> <span m="1360690">is</span> <span m="1361010">an</span> <span m="1361260">Euler tour.</span> <span m="1361560">So</span> <span m="1361710">now</span> <span m="1361880">we</span> <span m="1361980">have</span> <span m="1362440">shown</span> <span m="1362830">the</span> <span m="1362930">existence</span> <span m="1363330">of an</span> <span m="1363470">Euler</span> <span m="1363780">tour,</span> <span m="1364680">and</span> <span m="1364810">that</span> <span m="1364940">was the</span> <span m="1365290">other</span> <span m="1365360">direction</span> <span m="1365780">of</span> <span m="1365880">the</span> <span m="1365960">theorem.</span> </p>
<p><span m="1369800">So</span> <span m="1370670">let's</span> <span m="1371300">now</span> <span m="1371480">start</span> <span m="1371970">with</span> <span m="1372800">a</span> <span m="1372840">new</span> <span m="1373010">topic</span> <span m="1374000">on</span> <span m="1374400">directed</span> <span m="1374820">graphs.</span> <span m="1376130">And</span> <span m="1377960">later</span> <span m="1378270">on,</span> <span m="1378370">we</span> <span m="1378510">will</span> <span m="1378820">talk</span> <span m="1379370">about</span> <span m="1379720">how</span> <span m="1379900">Hamiltonian</span> <span m="1380320">paths.</span> <span m="1381140">And</span> <span m="1381340">these</span> <span m="1381650">are</span> <span m="1382350">different</span> <span m="1382810">from</span> <span m="1382980">an</span> <span m="1383110">Euler</span> <span m="1383390">tour</span> <span m="1384280">in that</span> <span m="1384880">in</span> <span m="1385220">an</span> <span m="1385380">Euler</span> <span m="1385780">tour,</span> <span m="1386470">every</span> <span m="1386820">edge</span> <span m="1387260">is</span> <span m="1387400">used</span> <span m="1387700">exactly</span> <span m="1388120">once.</span> <span m="1388490">In</span> <span m="1388620">a</span> <span m="1388650">Hamiltonian</span> <span m="1389200">path,</span> <span m="1390000">we</span> <span m="1390150">will</span> <span m="1390260">have</span> <span m="1390490">that</span> <span m="1390690">every</span> <span m="1390830">vertex</span> <span m="1391270">is</span> <span m="1391590">used</span> <span m="1391820">exactly</span> <span m="1392210">once.</span> <span m="1393160">So</span> <span m="1393320">we'll</span> <span m="1393790">do</span> <span m="1394200">that</span> <span m="1394600">a</span> <span m="1394640">bit</span> <span m="1394800">later.</span> </p>
<p><span m="1397100">So</span> <span m="1397250">what</span> <span m="1397410">are</span> <span m="1397470">directed</span> <span m="1397900">graphs?</span> <span m="1398660">Directed</span> <span m="1399060">graphs</span> <span m="1399410">are</span> <span m="1399480">graphs</span> <span m="1399920">where</span> <span m="1400210">we</span> <span m="1400390">have</span> <span m="1400930">edges</span> <span m="1402860">that</span> <span m="1402970">have</span> <span m="1403150">a</span> <span m="1403190">specific</span> <span m="1403650">direction.</span> <span m="1404410">So</span> <span m="1404850">we</span> <span m="1404980">can</span> <span m="1405200">walk</span> <span m="1405550">from</span> <span m="1407810">one</span> <span m="1407980">vertex</span> <span m="1408380">to</span> <span m="1408480">the</span> <span m="1408630">other.</span> <span m="1412760">Let</span> <span m="1413060">me</span> <span m="1413140">just</span> <span m="1413680">depict</span> <span m="1414040">an</span> <span m="1414110">example.</span> </p>
<p><span m="1418050">For</span> <span m="1418670">example,</span> <span m="1419180">we</span> <span m="1419340">could</span> <span m="1419560">have</span> <span m="1420100">something</span> <span m="1420500">that</span> <span m="1420650">looks</span> <span m="1420870">like</span> <span m="1421040">this.</span> <span m="1422420">We</span> <span m="1422440">have</span> <span m="1422680">V1.</span> <span m="1423960">We</span> <span m="1424100">have</span> <span m="1424850">V2,</span> <span m="1426600">V3.</span> <span m="1427950">We</span> <span m="1428020">may</span> <span m="1428180">have</span> <span m="1428540">an</span> <span m="1428660">edge that</span> <span m="1428930">goes</span> <span m="1429210">like</span> <span m="1429490">this</span> <span m="1430640">with</span> <span m="1430770">an</span> <span m="1430890">arrow.</span> <span m="1431360">We use</span> <span m="1431640">arrows</span> <span m="1432030">in</span> <span m="1432110">this</span> <span m="1432320">case.</span> <span m="1433720">I</span> <span m="1433740">can</span> <span m="1433920">go</span> <span m="1434120">to</span> <span m="1434300">V2</span> <span m="1434570">and</span> <span m="1434940">V3.</span> <span m="1435710">From</span> <span m="1436120">V2</span> <span m="1436360">I</span> <span m="1436540">may</span> <span m="1436810">be</span> <span m="1436970">able</span> <span m="1437090">to</span> <span m="1437210">go</span> <span m="1437390">to</span> <span m="1437520">V3,</span> <span m="1437830">and</span> <span m="1438270">from</span> <span m="1438480">V3</span> <span m="1438810">to</span> <span m="1439310">V2.</span> </p>
<p><span m="1440920">So</span> <span m="1441080">this</span> <span m="1441280">will</span> <span m="1441370">be</span> <span m="1441540">a</span> <span m="1441740">directed</span> <span m="1441960">graph,</span> <span m="1442430">as an</span> <span m="1442620">example.</span> <span m="1444100">We</span> <span m="1444220">also</span> <span m="1444470">call</span> <span m="1444770">these</span> <span m="1445250">digraphs.</span> <span m="1449680">And</span> <span m="1450310">as</span> <span m="1450450">you</span> <span m="1450570">can</span> <span m="1450750">see,</span> <span m="1451830">we</span> <span m="1451920">may</span> <span m="1452120">have</span> <span m="1452320">an</span> <span m="1452430">edge</span> <span m="1452800">that goes</span> <span m="1453060">from</span> <span m="1453140">V2</span> <span m="1453600">to</span> <span m="1454060">V3</span> <span m="1454820">and</span> <span m="1454980">also</span> <span m="1455250">one</span> <span m="1455440">that</span> <span m="1455570">goes</span> <span m="1455810">from</span> <span m="1455910">V3</span> <span m="1456160">to</span> <span m="1456860">V2.</span> <span m="1457330">So</span> <span m="1457460">these</span> <span m="1457660">are</span> <span m="1457740">two</span> <span m="1457930">separate</span> <span m="1458420">edges.</span> <span m="1459690">So</span> <span m="1459890">every edge</span> <span m="1460370">has</span> <span m="1460540">a</span> <span m="1460590">direction.</span> <span m="1461960">And</span> <span m="1463430">we</span> <span m="1463510">usually</span> <span m="1463930">say</span> <span m="1464320">that</span> <span m="1465270">if</span> <span m="1465420">we</span> <span m="1465510">have</span> <span m="1465750">an</span> <span m="1465870">edge</span> <span m="1466870">that</span> <span m="1467140">points</span> <span m="1467490">from</span> <span m="1467690">V2</span> <span m="1467890">to,</span> <span m="1468150">say,</span> <span m="1468553">V3,</span> <span m="1470370">then</span> <span m="1470780">we</span> <span m="1470900">call</span> <span m="1471160">this</span> <span m="1472640">the</span> <span m="1473540">tail</span> <span m="1475210">and</span> <span m="1475450">over</span> <span m="1475700">here</span> <span m="1475910">we</span> <span m="1476070">have</span> <span m="1476310">the</span> <span m="1476390">head.</span> <span m="1477530">So</span> <span m="1477690">this</span> <span m="1477880">is</span> <span m="1477980">some</span> <span m="1478140">notation</span> <span m="1478720">that</span> <span m="1478860">you</span> <span m="1478940">may</span> <span m="1479130">use.</span> </p>
<p><span m="1483840">We</span> <span m="1484000">also</span> <span m="1484280">have</span> <span m="1484510">now a</span> <span m="1484740">different</span> <span m="1485120">notion</span> <span m="1485520">for the</span> <span m="1485830">degree</span> <span m="1486290">of</span> <span m="1487000">a</span> <span m="1487300">vertex</span> <span m="1488240">because</span> <span m="1488640">we have</span> <span m="1488800">different</span> <span m="1489100">types</span> <span m="1489420">of</span> <span m="1489620">edges,</span> <span m="1490010">essentially.</span> <span m="1491260">Take,</span> <span m="1491590">for</span> <span m="1491690">example,</span> <span m="1491950">V2.</span> <span m="1493330">You</span> <span m="1493480">have</span> <span m="1493730">incoming</span> <span m="1494250">edges</span> <span m="1494730">and</span> <span m="1494870">outgoing</span> <span m="1495450">edges.</span> <span m="1496370">So</span> <span m="1496510">that's</span> <span m="1496710">why</span> <span m="1496930">we're</span> <span m="1497090">going</span> <span m="1497290">to</span> <span m="1497390">talk</span> <span m="1497610">about</span> <span m="1497840">an</span> <span m="1497950">indegree</span> <span m="1502220">and</span> <span m="1502520">an</span> <span m="1502630">outdegree.</span> <span m="1503120">So,</span> <span m="1503380">for example, the</span> <span m="1503650">indegree</span> <span m="1504150">of</span> <span m="1504250">V2</span> <span m="1506020">is</span> <span m="1506230">equal</span> <span m="1506540">to,</span> <span m="1506900">well,</span> <span m="1507110">I</span> <span m="1507210">have</span> <span m="1507410">one,</span> <span m="1508050">two</span> <span m="1508310">incoming</span> <span m="1508850">edges.</span> <span m="1510380">And</span> <span m="1510630">the</span> <span m="1511120">outdegree</span> <span m="1514200">of</span> <span m="1514410">V2</span> <span m="1515860">is</span> <span m="1516050">actually</span> <span m="1516300">something</span> <span m="1516640">different.</span> <span m="1517060">It's</span> <span m="1517270">just</span> <span m="1517530">one</span> <span m="1517790">outgoing</span> <span m="1518360">edge.</span> <span m="1518790">So</span> <span m="1518850">this</span> <span m="1519030">is</span> <span m="1519160">equal</span> <span m="1519530">to</span> <span m="1519710">1.</span> <span m="1522010">So</span> <span m="1522140">this</span> <span m="1522320">is</span> <span m="1522400">some</span> <span m="1522580">notation.</span> </p>
<p><span m="1524560">So</span> <span m="1524800">let's</span> <span m="1525470">talk</span> <span m="1525840">about</span> <span m="1527760">walks.</span> <span m="1528930">Let's</span> <span m="1529270">figure</span> <span m="1529640">out</span> <span m="1529780">where</span> <span m="1529890">we</span> <span m="1530020">can</span> <span m="1530210">enumerate</span> <span m="1530850">walks</span> <span m="1532020">in</span> <span m="1532430">a</span> <span m="1532520">directed</span> <span m="1532960">graph</span> <span m="1533980">and</span> <span m="1534930">compute</span> <span m="1535300">those.</span> <span m="1535940">So</span> <span m="1536030">how</span> <span m="1536190">many</span> <span m="1536460">walks</span> <span m="1536820">do</span> <span m="1537040">I have,</span> <span m="1538240">say,</span> <span m="1538830">of</span> <span m="1539060">length</span> <span m="1539450">k</span> <span m="1539870">that</span> <span m="1540110">go</span> <span m="1540260">exactly</span> <span m="1540830">from,</span> <span m="1541060">say,</span> <span m="1541400">V1</span> <span m="1541610">to</span> <span m="1542330">V3.</span> <span m="1543180">How</span> <span m="1543350">can</span> <span m="1543510">I</span> <span m="1543550">compute</span> <span m="1543940">this?</span> </p>
<p><span m="1545290">So</span> <span m="1545790">I'd</span> <span m="1545870">like</span> <span m="1546050">to</span> <span m="1546150">introduce</span> <span m="1547110">adjacency</span> <span m="1547690">matrices.</span> <span m="1548840">You've</span> <span m="1548980">probably</span> <span m="1549300">seen</span> <span m="1549530">them</span> <span m="1549660">before.</span> <span m="1550540">But</span> <span m="1550650">let's</span> <span m="1550860">go</span> <span m="1551090">over</span> <span m="1551210">this</span> <span m="1551390">once</span> <span m="1551650">more</span> <span m="1552010">because</span> <span m="1552340">induction</span> <span m="1553070">is</span> <span m="1553240">also</span> <span m="1553510">important.</span> <span m="1555470">So</span> <span m="1555690">let's</span> <span m="1555900">do</span> <span m="1556000">the</span> <span m="1556080">theorem.</span> <span m="1557420">So</span> <span m="1557590">the</span> <span m="1557700">theorem</span> <span m="1558130">is this.</span> <span m="1560110">Suppose</span> <span m="1560530">we</span> <span m="1560610">have</span> <span m="1560810">a graph.</span> <span m="1565560">And</span> <span m="1565750">suppose</span> <span m="1566130">it</span> <span m="1566210">has</span> <span m="1566450">end</span> <span m="1567240">nodes.</span> <span m="1576170">And</span> <span m="1576540">we</span> <span m="1578290">have</span> <span m="1578610">the</span> <span m="1579060">vertices</span> <span m="1579590">V1</span> <span m="1580200">up to</span> <span m="1580510">Vn.</span> <span m="1582250">And</span> <span m="1582540">now</span> <span m="1582930">we</span> <span m="1583160">let</span> <span m="1584830">the</span> <span m="1584930">matrix</span> <span m="1585970">that</span> <span m="1586230">contains</span> <span m="1587440">the</span> <span m="1587590">entries</span> <span m="1588800">aij</span> <span m="1591060">denote</span> <span m="1593650">the</span> <span m="1594070">adjacency</span> <span m="1594800">matrix</span> <span m="1603130">for</span> <span m="1603200">G.</span> </p>
<p><span m="1606860">And</span> <span m="1608160">what</span> <span m="1608290">does</span> <span m="1608390">it mean?</span> <span m="1609910">It's</span> <span m="1610070">actually</span> <span m="1610400">means</span> <span m="1610695">that</span> <span m="1610990">in</span> <span m="1611130">this</span> <span m="1611410">case,</span> <span m="1612550">we</span> <span m="1613640">say</span> <span m="1614240">that</span> <span m="1615350">aij</span> <span m="1616680">is</span> <span m="1616940">equal</span> <span m="1617780">to</span> <span m="1617980">1</span> <span m="1618402">if</span> <span m="1620880">we</span> <span m="1621050">actually</span> <span m="1621360">have an</span> <span m="1621700">edge</span> <span m="1622170">that</span> <span m="1622840">goes</span> <span m="1623200">from</span> <span m="1623410">Vi</span> <span m="1624470">to</span> <span m="1625080">Vj--</span> <span m="1626520">so</span> <span m="1626680">this</span> <span m="1626920">is</span> <span m="1627080">an</span> <span m="1627210">edge--</span> <span m="1630170">and</span> <span m="1630350">0</span> <span m="1631640">if</span> <span m="1631870">this</span> <span m="1632040">is</span> <span m="1632160">not</span> <span m="1632320">the</span> <span m="1632400">case.</span> <span m="1635000">So</span> <span m="1635140">this</span> <span m="1635320">is</span> <span m="1635450">the</span> <span m="1635580">adjacency</span> <span m="1636170">matrix.</span> </p>
<p><span m="1637630">And</span> <span m="1639180">now</span> <span m="1639350">we</span> <span m="1639450">can</span> <span m="1639620">state</span> <span m="1639960">something</span> <span m="1640500">about</span> <span m="1640840">the</span> <span m="1640910">number</span> <span m="1641160">of</span> <span m="1642000">directed</span> <span m="1642460">walks</span> <span m="1644580">in</span> <span m="1644920">a</span> <span m="1645450">directed</span> <span m="1646030">graph.</span> <span m="1648150">So</span> <span m="1649700">it turns</span> <span m="1649960">out</span> <span m="1650460">that</span> <span m="1650720">this is</span> <span m="1650950">easily</span> <span m="1651300">computed</span> <span m="1654580">by</span> <span m="1654690">taking</span> <span m="1655000">powers</span> <span m="1655450">of the</span> <span m="1655590">adjacency</span> <span m="1656130">matrix.</span> <span m="1667890">Let</span> <span m="1668690">aij</span> <span m="1670960">with</span> <span m="1671260">the</span> <span m="1671370">superscript</span> <span m="1671940">k</span> <span m="1673600">be</span> <span m="1673800">equal</span> <span m="1674100">to</span> <span m="1674250">the</span> <span m="1674360">number</span> <span m="1675750">of</span> <span m="1676230">directed</span> <span m="1676820">walks</span> <span m="1681320">of</span> <span m="1681570">length</span> <span m="1683240">k</span> <span m="1686130">that</span> <span m="1686310">go</span> <span m="1686550">from</span> <span m="1688290">Vi</span> <span m="1691030">to</span> <span m="1691470">Vj.</span> <span m="1693530">So</span> <span m="1694420">you</span> <span m="1694540">want</span> <span m="1694670">to</span> <span m="1694750">compute</span> <span m="1695150">this</span> <span m="1695500">and</span> <span m="1695590">it</span> <span m="1695700">turns</span> <span m="1696040">out</span> <span m="1698110">that--</span> <span m="1701240">wait a</span> <span m="1701520">minute.</span> <span m="1702000">So</span> <span m="1702290">what</span> <span m="1702380">did</span> <span m="1702570">I do</span> <span m="1702830">here?</span> <span m="1705140">So actually</span> <span m="1705460">let</span> <span m="1706490">p</span> <span m="1707950">kij</span> <span m="1708710">be</span> <span m="1708850">the</span> <span m="1708930">number</span> <span m="1709190">of</span> <span m="1709330">directed</span> <span m="1709740">walks</span> <span m="1710010">of</span> <span m="1710250">length</span> <span m="1711090">k</span> <span m="1711480">from</span> <span m="1711720">phi i</span> <span m="1712030">to</span> <span m="1712210">phi</span> <span m="1713210">j.</span> </p>
<p><span m="1714440">Then</span> <span m="1714660">what</span> <span m="1714810">you</span> <span m="1714900">can</span> <span m="1715040">show</span> <span m="1715400">is</span> <span m="1715560">if you</span> <span m="1716120">look</span> <span m="1716480">at</span> <span m="1717350">matrix</span> <span m="1718450">A</span> <span m="1718600">to</span> <span m="1718730">the</span> <span m="1718830">power</span> <span m="1719150">k,</span> <span m="1721580">this</span> <span m="1721780">actually</span> <span m="1723080">is</span> <span m="1723280">the</span> <span m="1723360">matrix</span> <span m="1723950">that</span> <span m="1724110">contains</span> <span m="1724780">all</span> <span m="1725930">these</span> <span m="1728480">numbers.</span> <span m="1730190">So</span> <span m="1730690">let me</span> <span m="1730880">give</span> <span m="1731050">a</span> <span m="1731110">few</span> <span m="1731290">examples.</span> <span m="1734710">So</span> <span m="1735410">let's</span> <span m="1735770">take</span> <span m="1736010">the</span> <span m="1736100">matrix</span> <span m="1736610">over</span> <span m="1736950">here.</span> <span m="1737670">Look</span> <span m="1738070">at</span> <span m="1738210">this</span> <span m="1738390">ajc</span> <span m="1738930">matrix,</span> <span m="1739460">take</span> <span m="1739660">a</span> <span m="1739760">few</span> <span m="1739930">powers</span> <span m="1740240">and</span> <span m="1740390">see</span> <span m="1740560">what</span> <span m="1740700">happens.</span> <span m="1742030">So</span> <span m="1742260">the</span> <span m="1742360">matrix</span> <span m="1742790">looks</span> <span m="1743000">like</span> <span m="1743190">this.</span> <span m="1744210">If</span> <span m="1744380">you just</span> <span m="1745840">label</span> <span m="1746310">the</span> <span m="1746430">rows</span> <span m="1746980">by,</span> <span m="1747330">say,</span> <span m="1747610">V1,</span> <span m="1748090">V2,</span> <span m="1748390">and</span> <span m="1748865">V3.</span> <span m="1749590">The</span> <span m="1749720">columns</span> <span m="1750100">by</span> <span m="1750250">V1,</span> <span m="1750790">V2,</span> <span m="1751400">and</span> <span m="1751550">V3.</span> <span m="1753340">Well,</span> <span m="1753700">V1</span> <span m="1753990">has</span> <span m="1754340">an</span> <span m="1754420">edge</span> <span m="1754680">to</span> <span m="1754850">V1.</span> <span m="1756020">V1</span> <span m="1756330">has</span> <span m="1756430">an</span> <span m="1756520">edge</span> <span m="1756710">to</span> <span m="1756820">V2</span> <span m="1757370">and</span> <span m="1757480">also</span> <span m="1757720">one</span> <span m="1757900">to</span> <span m="1758000">V3.</span> <span m="1759360">V2</span> <span m="1759550">has only</span> <span m="1759890">one</span> <span m="1760140">edge</span> <span m="1760490">to</span> <span m="1760790">V3,</span> <span m="1762710">so</span> <span m="1762970">we</span> <span m="1763070">have</span> <span m="1763190">zeros</span> <span m="1763630">here.</span> <span m="1764560">And</span> <span m="1764880">we</span> <span m="1764990">have</span> <span m="1765200">this.</span> <span m="1766710">So</span> <span m="1766860">this</span> <span m="1767060">is</span> <span m="1767200">the</span> <span m="1767270">matrix</span> <span m="1767810">A.</span> </p>
<p><span m="1770210">For</span> <span m="1770330">example,</span> <span m="1770760">if</span> <span m="1770950">you</span> <span m="1771050">compute</span> <span m="1771690">a</span> <span m="1771900">squared.</span> <span m="1773180">So</span> <span m="1773330">how</span> <span m="1773440">do</span> <span m="1773540">we</span> <span m="1773650">do</span> <span m="1773800">this?</span> <span m="1774510">How</span> <span m="1774620">many</span> <span m="1774850">of</span> <span m="1774980">you</span> <span m="1775130">know</span> <span m="1775370">about</span> <span m="1775630">matrix</span> <span m="1775950">multiplication?</span> <span m="1777210">Everybody</span> <span m="1777710">knows?</span> <span m="1781310">Well</span> <span m="1781660">anyway,</span> <span m="1781970">so</span> <span m="1782520">let's</span> <span m="1782750">assume</span> <span m="1783080">that</span> <span m="1783200">you</span> <span m="1783330">do</span> <span m="1783500">know</span> <span m="1783750">actually.</span> <span m="1787390">So</span> <span m="1787560">that's</span> <span m="1787760">pretty</span> <span m="1788000">simple.</span> <span m="1788560">So</span> <span m="1788640">you</span> <span m="1788780">take</span> <span m="1789030">the</span> <span m="1789140">column.</span> <span m="1789580">You</span> <span m="1789700">take</span> <span m="1789880">the</span> <span m="1789980">inners</span> <span m="1790170">product</span> <span m="1790520">with</span> <span m="1790650">the</span> <span m="1790720">first</span> <span m="1791090">row,</span> <span m="1791430">and</span> <span m="1791600">so</span> <span m="1791800">on.</span> <span m="1792110">And</span> <span m="1792250">you</span> <span m="1792360">can</span> <span m="1792820">easily</span> <span m="1793170">compute</span> <span m="1793590">this</span> <span m="1794610">otherwise</span> <span m="1794840">you</span> <span m="1794930">may</span> <span m="1795080">want</span> <span m="1795240">to</span> <span m="1795350">do</span> <span m="1795450">it</span> <span m="1796190">yourself</span> <span m="1796740">later</span> <span m="1797040">on.</span> <span m="1797810">And</span> <span m="1797960">you</span> <span m="1798050">get</span> <span m="1798270">this</span> <span m="1798500">particular</span> <span m="1800930">matrix.</span> <span m="1802270">And,</span> <span m="1802640">for</span> <span m="1802780">example,</span> <span m="1803740">a</span> <span m="1804220">to</span> <span m="1804460">the</span> <span m="1804540">power of</span> <span m="1804820">3</span> <span m="1805410">is</span> <span m="1805560">something</span> <span m="1805910">very</span> <span m="1806120">similar--</span> <span m="1807170">1,</span> <span m="1807637">3, 3,</span> <span m="1808571">0, 0,</span> <span m="1809038">1,</span> <span m="1810439">0,</span> <span m="1810906">1,</span> <span m="1811373">0.</span> </p>
<p><span m="1812900">Well,</span> <span m="1813610">it</span> <span m="1813790">turns</span> <span m="1814000">out</span> <span m="1814370">that</span> <span m="1814550">if</span> <span m="1814700">you</span> <span m="1814820">look</span> <span m="1814980">at</span> <span m="1815030">this</span> <span m="1815210">graph</span> <span m="1815700">and</span> <span m="1815850">we,</span> <span m="1815960">for</span> <span m="1816100">example,</span> <span m="1816430">want</span> <span m="1816610">to</span> <span m="1816710">know</span> <span m="1816900">the</span> <span m="1816990">number</span> <span m="1817380">of</span> <span m="1817580">walks</span> <span m="1817970">of</span> <span m="1818210">length</span> <span m="1819060">3</span> <span m="1819510">that</span> <span m="1819760">go</span> <span m="1819970">from</span> <span m="1820170">V1</span> <span m="1820770">to,</span> <span m="1820880">say,</span> <span m="1821160">V2,</span> <span m="1822380">let's</span> <span m="1823120">see</span> <span m="1823390">whether</span> <span m="1823570">we</span> <span m="1823700">can</span> <span m="1825470">see</span> <span m="1825770">those</span> <span m="1827260">over</span> <span m="1827570">here.</span> <span m="1829370">Well,</span> <span m="1830090">if</span> <span m="1830360">I</span> <span m="1830900">travel</span> <span m="1831230">from</span> <span m="1831430">V1</span> <span m="1831840">to</span> <span m="1832000">V2</span> <span m="1832290">in</span> <span m="1832800">three</span> <span m="1833080">steps,</span> <span m="1833830">I</span> <span m="1833990">can</span> <span m="1834270">go</span> <span m="1836030">like</span> <span m="1836290">this.</span> <span m="1836600">I</span> <span m="1836640">can</span> <span m="1836920">go</span> <span m="1837610">one,</span> <span m="1838400">two,</span> <span m="1838890">three.</span> <span m="1839350">So</span> <span m="1839480">that's</span> <span m="1839650">one</span> <span m="1839850">option.</span> <span m="1841010">I</span> <span m="1841170">can</span> <span m="1841410">go</span> <span m="1842160">one,</span> <span m="1842910">and</span> <span m="1843120">another</span> <span m="1843490">one</span> <span m="1843870">time,</span> <span m="1844240">so it's a</span> <span m="1844510">second</span> <span m="1844910">step,</span> <span m="1845380">and</span> <span m="1845610">three.</span> <span m="1846470">I</span> <span m="1846550">can</span> <span m="1846740">also</span> <span m="1846940">go</span> <span m="1847120">directly</span> <span m="1847570">over</span> <span m="1847860">here</span> <span m="1848230">and</span> <span m="1848420">go</span> <span m="1848620">back</span> <span m="1848880">here,</span> <span m="1849200">and</span> <span m="1849440">back</span> <span m="1849650">over</span> <span m="1849920">here.</span> </p>
<p><span m="1850960">So</span> <span m="1851170">it</span> <span m="1851260">turns</span> <span m="1851570">out</span> <span m="1851850">that's</span> <span m="1852690">I</span> <span m="1852790">can</span> <span m="1852990">compute</span> <span m="1853390">this.</span> <span m="1853620">So</span> <span m="1853740">how</span> <span m="1853880">do</span> <span m="1853980">we</span> <span m="1854110">prove</span> <span m="1854460">this?</span> <span m="1855580">We'll</span> <span m="1855760">use</span> <span m="1856850">induction.</span> <span m="1858370">And</span> <span m="1859320">the</span> <span m="1861140">steps</span> <span m="1862720">are</span> <span m="1862880">pretty</span> <span m="1863180">straightforward.</span> </p>
<p><span m="1866130">So</span> <span m="1866240">let's</span> <span m="1866480">first</span> <span m="1866730">define,</span> <span m="1868650">because</span> <span m="1868930">we</span> <span m="1869020">want</span> <span m="1869220">to</span> <span m="1869280">prove</span> <span m="1869720">that</span> <span m="1870360">kth</span> <span m="1870710">power</span> <span m="1871060">of</span> <span m="1871230">a</span> <span m="1871610">is</span> <span m="1871750">equal</span> <span m="1872000">to</span> <span m="1872090">that</span> <span m="1872230">matrix</span> <span m="1872730">up</span> <span m="1872860">there,</span> <span m="1873670">where</span> <span m="1875660">all</span> <span m="1875850">the</span> <span m="1876000">entries</span> <span m="1876470">represent</span> <span m="1877630">the</span> <span m="1877710">number</span> <span m="1877980">of</span> <span m="1878090">walks.</span> <span m="1879350">So</span> <span m="1879620">let's</span> <span m="1880600">say</span> <span m="1881250">that</span> <span m="1882780">aijk,</span> <span m="1885310">that this</span> <span m="1885710">denote</span> <span m="1888750">the</span> <span m="1889650">i</span> <span m="1890870">jth</span> <span m="1892690">entry</span> <span m="1895530">in</span> <span m="1897010">a</span> <span m="1897740">to</span> <span m="1897880">the</span> <span m="1897980">power</span> <span m="1898290">k.</span> <span m="1902610">And</span> <span m="1902850">you</span> <span m="1902930">want</span> <span m="1903080">to</span> <span m="1903170">show</span> <span m="1903950">that</span> <span m="1904480">this</span> <span m="1904820">number</span> <span m="1905080">is</span> <span m="1905200">actually</span> <span m="1905510">equal</span> <span m="1906510">to</span> <span m="1906820">pijk</span> <span m="1908170">for</span> <span m="1910280">all</span> <span m="1910420">the</span> <span m="1910540">entries.</span> </p>
<p><span m="1911930">So</span> <span m="1913190">if</span> <span m="1913380">you're</span> <span m="1913550">going</span> <span m="1913730">to</span> <span m="1913860">use</span> <span m="1914070">induction,</span> <span m="1915620">it</span> <span m="1915720">makes</span> <span m="1916010">sense</span> <span m="1916410">to</span> <span m="1918100">assume</span> <span m="1919300">that</span> <span m="1919730">the</span> <span m="1919830">theorem</span> <span m="1920320">is</span> <span m="1920620">true</span> <span m="1920970">for</span> <span m="1921620">k.</span> <span m="1923440">So</span> <span m="1924826">the</span> <span m="1925290">induction</span> <span m="1925830">hypothesis</span> <span m="1927000">could</span> <span m="1927170">be</span> <span m="1927270">something</span> <span m="1927620">like</span> <span m="1927790">this.</span> <span m="1930000">The</span> <span m="1930120">theorem</span> <span m="1932270">is</span> <span m="1932560">true</span> <span m="1934980">for</span> <span m="1935350">k.</span> <span m="1936700">And</span> <span m="1936900">this</span> <span m="1937080">is</span> <span m="1937250">really</span> <span m="1937480">the</span> <span m="1937590">same</span> <span m="1938320">as</span> <span m="1939360">stating</span> <span m="1940900">that</span> <span m="1942480">all</span> <span m="1942730">these</span> <span m="1943170">entries</span> <span m="1944190">for</span> <span m="1944460">all</span> <span m="1947630">i</span> <span m="1948030">and</span> <span m="1948270">j--</span> <span m="1951140">aij</span> <span m="1952370">is</span> <span m="1952600">equal</span> <span m="1952870">to</span> <span m="1953340">the</span> <span m="1953774">pijk.</span> <span m="1957310">So</span> <span m="1957470">this</span> <span m="1957730">is</span> <span m="1958090">what</span> <span m="1958240">you</span> <span m="1958380">want</span> <span m="1958520">to</span> <span m="1958620">prove.</span> <span m="1959710">And</span> <span m="1960500">this</span> <span m="1960660">is</span> <span m="1960760">pretty</span> <span m="1960970">straightforward</span> <span m="1961780">because</span> <span m="1962060">now</span> <span m="1962370">what</span> <span m="1962480">we</span> <span m="1962560">can</span> <span m="1962750">do</span> <span m="1963000">is</span> <span m="1963980">we</span> <span m="1964060">can</span> <span m="1964220">start</span> <span m="1964780">to</span> <span m="1967710">look</span> <span m="1968060">at</span> <span m="1968560">how</span> <span m="1968880">we</span> <span m="1969000">compute</span> <span m="1969730">walks</span> <span m="1970100">of</span> <span m="1970230">length</span> <span m="1970580">k,</span> <span m="1971100">or</span> <span m="1971300">k</span> <span m="1971490">plus</span> <span m="1971760">1.</span> </p>
<p><span m="1972020">So</span> <span m="1972170">let's</span> <span m="1972370">first</span> <span m="1972600">do</span> <span m="1973060">the</span> <span m="1973370">base</span> <span m="1973720">case.</span> <span m="1974110">That's</span> <span m="1974250">how</span> <span m="1974380">we</span> <span m="1974470">always</span> <span m="1974770">start.</span> <span m="1976680">The</span> <span m="1976780">base</span> <span m="1977120">case</span> <span m="1977470">is</span> <span m="1977690">k</span> <span m="1978000">equals</span> <span m="1978390">1.</span> <span m="1979210">And</span> <span m="1981070">we</span> <span m="1981210">have</span> <span m="1981440">essentially</span> <span m="1982120">two</span> <span m="1982610">options.</span> <span m="1983850">So</span> <span m="1984070">we</span> <span m="1984190">want</span> <span m="1984330">to</span> <span m="1984420">prove</span> <span m="1984680">this,</span> <span m="1984850">so take</span> <span m="1984970">an</span> <span m="1985210">i</span> <span m="1985500">and a</span> <span m="1985730">j,</span> <span m="1986560">whatever</span> <span m="1986930">you</span> <span m="1987070">want.</span> <span m="1988340">So</span> <span m="1988590">suppose</span> <span m="1989070">we</span> <span m="1989160">has</span> <span m="1989430">an</span> <span m="1989570">edge,</span> <span m="1990170">Vi,</span> <span m="1991450">that</span> <span m="1991650">goes</span> <span m="1992020">to</span> <span m="1992470">Vj.</span> <span m="1994030">Well</span> <span m="1994720">in</span> <span m="1994880">that</span> <span m="1995070">case,</span> <span m="1995680">we</span> <span m="1995880">know</span> <span m="1996140">that</span> <span m="1996410">the</span> <span m="1996490">number</span> <span m="1996820">of</span> <span m="1997020">walks</span> <span m="1998370">of</span> <span m="1998610">length</span> <span m="1998910">1,</span> <span m="2002330">from</span> <span m="2002950">i</span> <span m="2003180">to</span> <span m="2003320">j,</span> <span m="2003700">is exactly</span> <span m="2004190">1</span> <span m="2004400">because</span> <span m="2004660">there's</span> <span m="2004850">this</span> <span m="2005160">edge.</span> <span m="2006170">So</span> <span m="2006350">this</span> <span m="2006520">is</span> <span m="2006640">equal</span> <span m="2006880">to</span> <span m="2007050">1.</span> <span m="2007510">But</span> <span m="2007970">this is</span> <span m="2008210">also</span> <span m="2008460">the</span> <span m="2008580">definition</span> <span m="2009210">of</span> <span m="2009320">my</span> <span m="2009490">adjacency</span> <span m="2010110">matrix.</span> <span m="2011270">So</span> <span m="2011460">I</span> <span m="2011560">can</span> <span m="2011670">just</span> <span m="2011960">write</span> <span m="2012170">out</span> <span m="2012420">here</span> <span m="2013120">aij</span> <span m="2013270">1.</span> <span m="2015480">So</span> <span m="2015650">that's</span> <span m="2015880">great.</span> <span m="2016380">This</span> <span m="2016630">case</span> <span m="2017730">definitely</span> <span m="2018290">works.</span> </p>
<p><span m="2019780">Now,</span> <span m="2019870">if</span> <span m="2020050">there</span> <span m="2020230">is</span> <span m="2020350">no</span> <span m="2021190">edge</span> <span m="2022840">of</span> <span m="2023220">this</span> <span m="2023500">type,</span> <span m="2025530">then</span> <span m="2025760">you</span> <span m="2025880">know</span> <span m="2026350">that in</span> <span m="2026530">one</span> <span m="2026760">step,</span> <span m="2027410">we</span> <span m="2027550">can</span> <span m="2027910">never</span> <span m="2028130">achieve</span> <span m="2029070">Vj</span> <span m="2029580">from</span> <span m="2029790">Vi</span> <span m="2030210">because</span> <span m="2030420">there's</span> <span m="2030570">no</span> <span m="2030750">edge.</span> <span m="2031630">So</span> <span m="2031800">we</span> <span m="2031890">know this</span> <span m="2032230">is</span> <span m="2032340">equal</span> <span m="2032550">to</span> <span m="2032630">0.</span> <span m="2034420">There's</span> <span m="2034610">no</span> <span m="2034850">such</span> <span m="2035140">walk.</span> <span m="2036050">And</span> <span m="2036230">this</span> <span m="2036410">is,</span> <span m="2036540">by</span> <span m="2036680">definition</span> <span m="2037210">of the</span> <span m="2037310">adjacency</span> <span m="2037940">matrix,</span> <span m="2038490">also</span> <span m="2038850">equal</span> <span m="2039660">to</span> <span m="2040030">the</span> <span m="2040120">aij.</span> <span m="2041045">So</span> <span m="2041310">this</span> <span m="2041510">works.</span> <span m="2041860">So</span> <span m="2041960">the</span> <span m="2042000">base</span> <span m="2042290">case</span> <span m="2042540">it's</span> <span m="2042650">easy.</span> </p>
<p><span m="2044000">And</span> <span m="2045130">induction</span> <span m="2045610">step</span> <span m="2049190">always</span> <span m="2049510">starts</span> <span m="2049940">by</span> <span m="2050100">issuing</span> <span m="2051170">pk.</span> <span m="2056429">As</span> <span m="2056610">you</span> <span m="2056730">can</span> <span m="2056889">see,</span> <span m="2057670">these</span> <span m="2057969">types</span> <span m="2058150">of</span> <span m="2058489">proofs</span> <span m="2058570">always</span> <span m="2058830">have</span> <span m="2058969">the</span> <span m="2059060">same</span> <span m="2059330">structure.</span> <span m="2066340">In</span> <span m="2066469">this</span> <span m="2066750">case,</span> <span m="2068125">let</span> <span m="2068570">me again</span> <span m="2068870">assume</span> <span m="2073330">pk.</span> <span m="2079300">We</span> <span m="2079440">want</span> <span m="2079620">to</span> <span m="2080150">prove</span> <span m="2080850">pk</span> <span m="2081250">plus</span> <span m="2081460">1.</span> <span m="2082300">So</span> <span m="2082500">what</span> <span m="2082650">you</span> <span m="2082800">want</span> <span m="2082960">to</span> <span m="2083070">know</span> <span m="2083340">is</span> <span m="2083469">what's</span> <span m="2083770">the</span> <span m="2083870">number</span> <span m="2084080">of walks of</span> <span m="2084550">length</span> <span m="2084840">k</span> <span m="2085050">plus</span> <span m="2085400">1.</span> <span m="2086050">So</span> <span m="2086210">how</span> <span m="2086370">can</span> <span m="2086540">we</span> <span m="2087250">express</span> <span m="2087739">those?</span> <span m="2089110">pij</span> <span m="2090770">k</span> <span m="2091730">plus</span> <span m="2092070">1.</span> <span m="2093360">How</span> <span m="2094610">can</span> <span m="2094820">we</span> <span m="2095179">use</span> <span m="2095860">this</span> <span m="2096630">assumption</span> <span m="2097170">over</span> <span m="2097420">here?</span> <span m="2098260">Do you have</span> <span m="2098500">an</span> <span m="2098590">idea</span> <span m="2099900">so</span> <span m="2100140">we</span> <span m="2100260">can--</span> </p>
<p><span m="2104790">AUDIENCE: Any</span> <span m="2105710">walk</span> <span m="2106155">of</span> <span m="2106600">length</span> <span m="2107062">k plus</span> <span m="2107524">1</span> <span m="2108450">can be</span> <span m="2108590">got</span> <span m="2108870">by</span> <span m="2111560">taking</span> <span m="2112520">one</span> <span m="2113190">of--</span> <span m="2113680">let's</span> <span m="2114170">say</span> <span m="2114660">from</span> <span m="2115150">Vs</span> <span m="2115505">to</span> <span m="2115860">Vx.</span> <span m="2116356">[INAUDIBLE]</span> <span m="2118836">You go</span> <span m="2119828">from</span> <span m="2120330">Vs</span> <span m="2120750">to</span> <span m="2121105">V</span> <span m="2123236">in</span> <span m="2123680">k</span> <span m="2123940">steps</span> <span m="2125360">and</span> <span m="2125560">then</span> <span m="2125790">V to</span> <span m="2126220">Vf</span> <span m="2126620">in one</span> <span m="2126890">step.</span> </p>
<p><span m="2127810">PROFESSOR: That's true.</span> <span m="2128270">We</span> <span m="2128730">could</span> <span m="2129650">do that.</span> <span m="2130570">So</span> <span m="2130880">let's</span> <span m="2131350">write</span> <span m="2131520">it</span> <span m="2131660">down.</span> <span m="2131960">So</span> <span m="2132570">what you're</span> <span m="2132780">essentially</span> <span m="2133270">saying</span> <span m="2133710">is</span> <span m="2133860">that</span> <span m="2134070">you</span> <span m="2134320">can</span> <span m="2135310">enumerate</span> <span m="2135800">all</span> <span m="2135970">the</span> <span m="2136070">walks</span> <span m="2137150">by</span> <span m="2137570">going</span> <span m="2138720">of</span> <span m="2138940">length</span> <span m="2139400">k plus</span> <span m="2139580">1</span> <span m="2139710">by</span> <span m="2139830">first</span> <span m="2140160">going</span> <span m="2140390">from</span> <span m="2140700">i</span> <span m="2141870">in,</span> <span m="2142805">say,</span> <span m="2143390">k</span> <span m="2144140">steps,</span> <span m="2144830">to</span> <span m="2145700">whatever</span> <span m="2146380">V,</span> <span m="2147900">and</span> <span m="2148240">then</span> <span m="2149240">in</span> <span m="2149420">one</span> <span m="2149620">step</span> <span m="2150290">to,</span> <span m="2151020">say,</span> <span m="2151220">Vi</span> <span m="2151650">to</span> <span m="2152100">V</span> <span m="2152530">in</span> <span m="2152690">k</span> <span m="2152870">steps</span> <span m="2153170">and</span> <span m="2153350">then</span> <span m="2153690">in</span> <span m="2153840">one</span> <span m="2154030">step</span> <span m="2154420">to</span> <span m="2155334">Vj.</span> <span m="2158320">So</span> <span m="2158530">let's</span> <span m="2158750">write</span> <span m="2158970">this</span> <span m="2159180">down</span> <span m="2159460">because</span> <span m="2159790">V</span> <span m="2159950">can</span> <span m="2160080">be</span> <span m="2160250">anything.</span> <span m="2161310">So</span> <span m="2161370">what</span> <span m="2161470">to</span> <span m="2161590">do</span> <span m="2161780">is</span> <span m="2161910">we</span> <span m="2162030">have</span> <span m="2162220">a</span> <span m="2162260">sum</span> <span m="2163720">over</span> <span m="2164030">all</span> <span m="2164270">the</span> <span m="2164370">Vs</span> <span m="2165320">such</span> <span m="2165740">that--</span> <span m="2166410">well,</span> <span m="2166900">we</span> <span m="2167040">will</span> <span m="2167370">use</span> <span m="2167770">indices</span> <span m="2168260">here</span> <span m="2168600">let me</span> <span m="2168840">do that</span> <span m="2168940">a</span> <span m="2169600">little</span> <span m="2169780">bit</span> <span m="2169910">differently.</span> <span m="2171030">So</span> <span m="2171220">let's</span> <span m="2171460">say</span> <span m="2171590">I</span> <span m="2171840">have an</span> <span m="2171980">h</span> <span m="2172300">over</span> <span m="2172550">here.</span> <span m="2172800">So</span> <span m="2172980">all</span> <span m="2174750">the</span> <span m="2174830">indices</span> <span m="2175360">h</span> <span m="2176260">such</span> <span m="2176610">that</span> <span m="2177230">Vh</span> <span m="2177830">to</span> <span m="2178702">Vj</span> <span m="2180110">is</span> <span m="2180470">actually</span> <span m="2181550">an</span> <span m="2181770">edge</span> <span m="2183160">in</span> <span m="2183930">G.</span> </p>
<p><span m="2186110">So</span> <span m="2186320">then</span> <span m="2186480">we</span> <span m="2186580">can</span> <span m="2186730">write</span> <span m="2186990">out</span> <span m="2187200">here</span> <span m="2187840">that</span> <span m="2188050">we</span> <span m="2190360">go in</span> <span m="2190630">k</span> <span m="2190840">steps</span> <span m="2191240">to</span> <span m="2191740">Vh.</span> <span m="2192020">So</span> <span m="2192310">how</span> <span m="2193690">many</span> <span m="2194070">walks</span> <span m="2194540">are</span> <span m="2194800">there?</span> <span m="2195110">Well,</span> <span m="2195390">we</span> <span m="2195540">can use</span> <span m="2195700">the induction</span> <span m="2196060">hypothesis</span> <span m="2196580">now,</span> <span m="2196800">right?</span> <span m="2197610">So</span> <span m="2197800">we</span> <span m="2197920">can</span> <span m="2198120">say</span> <span m="2198860">you</span> <span m="2198950">go</span> <span m="2199160">from</span> <span m="2199460">i</span> <span m="2199760">to</span> <span m="2200170">h</span> <span m="2201330">in</span> <span m="2201520">k</span> <span m="2201760">steps.</span> <span m="2203130">And</span> <span m="2203440">then,</span> <span m="2205970">well,</span> <span m="2207610">we</span> <span m="2207750">can</span> <span m="2207900">use</span> <span m="2208150">this</span> <span m="2208300">particular</span> <span m="2208730">edge</span> <span m="2209360">to</span> <span m="2209910">complete</span> <span m="2210400">it</span> <span m="2211090">to</span> <span m="2211420">a</span> <span m="2211550">k plus</span> <span m="2212020">1th</span> <span m="2212360">walk</span> <span m="2212990">from</span> <span m="2213290">i</span> <span m="2213560">to</span> <span m="2213790">j.</span> </p>
<p><span m="2215960">So</span> <span m="2216130">now</span> <span m="2216620">what</span> <span m="2216810">is</span> <span m="2216940">this</span> <span m="2217130">equal</span> <span m="2217470">to?</span> <span m="2218110">Can</span> <span m="2218270">we</span> <span m="2218380">simplify</span> <span m="2218970">this</span> <span m="2219290">sum?</span> <span m="2220550">We</span> <span m="2220650">can,</span> <span m="2220910">right,</span> <span m="2221260">because</span> <span m="2223520">we</span> <span m="2223660">know</span> <span m="2223920">that</span> <span m="2224330">there's</span> <span m="2224830">an</span> <span m="2224960">edge</span> <span m="2225770">if</span> <span m="2226010">and</span> <span m="2226160">only</span> <span m="2226590">if</span> <span m="2227690">the</span> <span m="2227800">adjacency</span> <span m="2228370">matrix</span> <span m="2228820">has</span> <span m="2228960">a</span> <span m="2229080">1</span> <span m="2229310">in</span> <span m="2229430">a</span> <span m="2229510">particular</span> <span m="2229890">position.</span> <span m="2231300">So</span> <span m="2231440">we</span> <span m="2231570">could</span> <span m="2232340">rewrite</span> <span m="2232890">it</span> <span m="2233220">and</span> <span m="2233420">sum</span> <span m="2233690">over</span> <span m="2234060">all</span> <span m="2234470">h</span> <span m="2236430">from</span> <span m="2236730">1</span> <span m="2237000">to</span> <span m="2237980">n.</span> <span m="2239910">And</span> <span m="2241500">write</span> <span m="2241950">pij</span> <span m="2243714">k</span> <span m="2244600">times,</span> <span m="2245900">and</span> <span m="2246090">then</span> <span m="2246930">aj--</span> <span m="2249060">oh,</span> <span m="2249970">this should</span> <span m="2250230">be an</span> <span m="2250360">h.</span> <span m="2252510">So</span> <span m="2252640">we</span> <span m="2252740">have</span> <span m="2252840">the</span> <span m="2252940">same</span> <span m="2253160">over</span> <span m="2253520">here.</span> <span m="2254100">And</span> <span m="2254240">then</span> <span m="2254390">we</span> <span m="2254610">have one</span> <span m="2254970">edge</span> <span m="2255360">from</span> <span m="2255660">h</span> <span m="2256220">to</span> <span m="2258770">2j.</span> </p>
<p><span m="2259610">So</span> <span m="2260540">I</span> <span m="2260750">only</span> <span m="2261150">count</span> <span m="2261640">this</span> <span m="2262240">number</span> <span m="2262590">over</span> <span m="2262870">here</span> <span m="2263060">if</span> <span m="2263230">this</span> <span m="2263420">is</span> <span m="2263530">equal</span> <span m="2263780">to</span> <span m="2263890">1,</span> <span m="2264360">and</span> <span m="2264480">that</span> <span m="2264600">happens</span> <span m="2264880">exactly</span> <span m="2265280">if</span> <span m="2265410">there's</span> <span m="2265660">an</span> <span m="2265760">edge.</span> <span m="2266500">I</span> <span m="2266610">do</span> <span m="2266720">not</span> <span m="2266990">count</span> <span m="2267290">this</span> <span m="2267800">if</span> <span m="2268010">there's</span> <span m="2268250">a</span> <span m="2268290">0</span> <span m="2268820">over</span> <span m="2268970">here,</span> <span m="2269230">that is</span> <span m="2269300">as</span> <span m="2269450">if</span> <span m="2269580">there's</span> <span m="2269800">no</span> <span m="2270040">edge.</span> <span m="2271930">But</span> <span m="2271990">now</span> <span m="2272570">we</span> <span m="2272840">can</span> <span m="2274490">use</span> <span m="2274830">induction</span> <span m="2275350">hypothesis</span> <span m="2275980">because</span> <span m="2276320">I</span> <span m="2276360">know that</span> <span m="2276650">these</span> <span m="2276900">numbers</span> <span m="2277690">are</span> <span m="2277860">equal</span> <span m="2278130">to</span> <span m="2278230">the</span> <span m="2278370">a's.</span> <span m="2279620">So</span> <span m="2279840">we</span> <span m="2280010">rewrite</span> <span m="2280440">this.</span> <span m="2283040">And</span> <span m="2283300">we</span> <span m="2283370">see</span> <span m="2283800">that</span> <span m="2283910">we get</span> <span m="2285150">aih</span> <span m="2285954">k.</span> <span m="2287760">So</span> <span m="2287920">this</span> <span m="2288130">is</span> <span m="2288630">where</span> <span m="2289050">we</span> <span m="2289190">use the</span> <span m="2289450">induction</span> <span m="2291210">step.</span> <span m="2291920">So</span> <span m="2292090">it's</span> <span m="2292320">like</span> <span m="2293930">we</span> <span m="2294030">assume</span> <span m="2294810">pk</span> <span m="2295450">over</span> <span m="2295720">here.</span> <span m="2298110">And</span> <span m="2298260">I</span> <span m="2298280">need</span> <span m="2298450">to</span> <span m="2298550">finish</span> <span m="2298860">this</span> <span m="2299010">formula.</span> <span m="2300530">So</span> <span m="2300650">we</span> <span m="2300750">have</span> <span m="2300930">this.</span> </p>
<p><span m="2301970">Now,</span> <span m="2303000">by</span> <span m="2303100">the</span> <span m="2303190">definition</span> <span m="2303650">of</span> <span m="2303810">matrix</span> <span m="2304150">multiplication,</span> <span m="2309400">we</span> <span m="2309580">actually</span> <span m="2310980">see</span> <span m="2311540">that</span> <span m="2311840">this</span> <span m="2312080">is</span> <span m="2312270">equal</span> <span m="2313510">to</span> <span m="2317710">a</span> <span m="2318900">k</span> <span m="2319150">plus</span> <span m="2319570">1</span> <span m="2320980">ij</span> <span m="2321700">to</span> <span m="2321960">the</span> <span m="2322190">ij-th</span> <span m="2322840">entry</span> <span m="2323840">in the</span> <span m="2324110">k plus oneth</span> <span m="2324910">power</span> <span m="2325390">of the</span> <span m="2325570">adjacency</span> <span m="2326020">matrix.</span> <span m="2327050">Here</span> <span m="2327290">we</span> <span m="2327380">have</span> <span m="2327830">this</span> <span m="2328000">represents</span> <span m="2328530">the</span> <span m="2328600">matrix</span> <span m="2329090">of</span> <span m="2329240">the</span> <span m="2329380">kth</span> <span m="2329700">power.</span> <span m="2330630">This</span> <span m="2330790">represents</span> <span m="2331360">the</span> <span m="2331410">matrix</span> <span m="2331850">of</span> <span m="2332570">a.</span> <span m="2333440">So</span> <span m="2333590">we</span> <span m="2333690">multiply</span> <span m="2334200">essentially</span> <span m="2334720">a</span> <span m="2334870">to</span> <span m="2334990">the</span> <span m="2335060">power</span> <span m="2335320">k</span> <span m="2335530">times</span> <span m="2335970">a</span> <span m="2336140">and get</span> <span m="2336450">a to</span> <span m="2336710">the</span> <span m="2336810">power</span> <span m="2337040">k</span> <span m="2337220">plus</span> <span m="2337520">1.</span> <span m="2338190">And</span> <span m="2338400">that's</span> <span m="2338610">what</span> <span m="2338790">we</span> <span m="2338880">see</span> <span m="2339110">here.</span> <span m="2340260">So</span> <span m="2340340">this</span> <span m="2340540">is</span> <span m="2340630">the</span> <span m="2340740">induction</span> <span m="2341120">step,</span> <span m="2341860">and</span> <span m="2342040">so</span> <span m="2342200">this</span> <span m="2342400">proves</span> <span m="2343440">the</span> <span m="2343570">theorem</span> <span m="2344020">up</span> <span m="2344150">here.</span> </p>
<p><span m="2347480">So</span> <span m="2347670">let's</span> <span m="2348920">talk</span> <span m="2349160">about</span> <span m="2349400">a</span> <span m="2349450">few</span> <span m="2349650">more</span> <span m="2349860">definitions</span> <span m="2351230">concerning</span> <span m="2351670">directed</span> <span m="2352090">graphs.</span> <span m="2353150">And</span> <span m="2354180">then</span> <span m="2354480">you</span> <span m="2354600">go</span> <span m="2355090">a</span> <span m="2355170">step</span> <span m="2355540">into</span> <span m="2357350">a</span> <span m="2357430">very</span> <span m="2357700">special</span> <span m="2358180">type</span> <span m="2358730">of</span> <span m="2360280">graph,</span> <span m="2361280">which</span> <span m="2361880">are</span> <span m="2362010">the</span> <span m="2362120">tournament</span> <span m="2362470">graphs</span> <span m="2362880">up</span> <span m="2362990">here.</span> <span m="2371840">One</span> <span m="2371990">of</span> <span m="2372110">the</span> <span m="2372210">things</span> <span m="2372460">that we</span> <span m="2372810">have</span> <span m="2373260">in</span> <span m="2373620">undirected</span> <span m="2374240">graphs,</span> <span m="2374570">so</span> <span m="2374920">where</span> <span m="2375190">we</span> <span m="2375310">have</span> <span m="2375540">no</span> <span m="2375780">directed</span> <span m="2376280">edges,</span> <span m="2377750">we</span> <span m="2377880">talked</span> <span m="2378400">last</span> <span m="2378760">time</span> <span m="2379040">a</span> <span m="2379110">lot</span> <span m="2379320">about</span> <span m="2380820">cyclicity</span> <span m="2381880">and</span> <span m="2382010">stuff</span> <span m="2382270">like</span> <span m="2382450">that.</span> <span m="2382720">And</span> <span m="2382840">we</span> <span m="2382930">talked</span> <span m="2383240">about</span> <span m="2384650">acyclic</span> <span m="2385410">connected</span> <span m="2386500">graphs.</span> <span m="2387530">And</span> <span m="2388140">as</span> <span m="2388600">we</span> <span m="2388730">categorized</span> <span m="2389500">those,</span> <span m="2389670">so</span> <span m="2389810">we</span> <span m="2389920">defined</span> <span m="2390360">them</span> <span m="2390600">as</span> <span m="2390840">trees,</span> <span m="2391350">and</span> <span m="2391490">they have</span> <span m="2391620">a</span> <span m="2391700">very</span> <span m="2391950">special</span> <span m="2392310">structure.</span> </p>
<p><span m="2394130">So</span> <span m="2396590">what</span> <span m="2396780">would</span> <span m="2396900">happen</span> <span m="2397230">here,</span> <span m="2397810">if</span> <span m="2397990">we</span> <span m="2398980">look</span> <span m="2399360">at</span> <span m="2403900">a</span> <span m="2403950">directed</span> <span m="2404450">graph,</span> <span m="2404950">and</span> <span m="2405090">we</span> <span m="2405220">wonder,</span> <span m="2406270">what</span> <span m="2406300">does</span> <span m="2406480">it</span> <span m="2406590">mean</span> <span m="2406800">to</span> <span m="2406910">be</span> <span m="2407060">connected.</span> <span m="2408260">Can we really</span> <span m="2408750">talk</span> <span m="2409070">about</span> <span m="2409320">that?</span> <span m="2410260">What</span> <span m="2410470">does</span> <span m="2410570">that mean?</span> <span m="2414350">For</span> <span m="2414510">example,</span> <span m="2415680">if</span> <span m="2415890">I</span> <span m="2415960">look</span> <span m="2416520">at</span> <span m="2416690">this</span> <span m="2416830">particular</span> <span m="2417260">example</span> <span m="2417720">graph</span> <span m="2418040">over</span> <span m="2418290">here,</span> <span m="2418900">I</span> <span m="2419030">can</span> <span m="2419270">say,</span> <span m="2419580">well,</span> <span m="2420220">V1</span> <span m="2420800">has</span> <span m="2421280">an</span> <span m="2421470">edge</span> <span m="2422080">that</span> <span m="2422320">points</span> <span m="2422730">towards</span> <span m="2422970">V1</span> <span m="2423640">itself,</span> <span m="2424200">and</span> <span m="2424500">towards</span> <span m="2424650">V2,</span> <span m="2425360">and</span> <span m="2425530">towards</span> <span m="2425660">V3.</span> <span m="2428220">So</span> <span m="2428700">maybe</span> <span m="2429060">I</span> <span m="2429120">would</span> <span m="2429310">call</span> <span m="2429550">this</span> <span m="2429710">graph</span> <span m="2430010">connected.</span> </p>
<p><span m="2431230">But</span> <span m="2431360">if</span> <span m="2431540">I</span> <span m="2431770">look</span> <span m="2432050">at</span> <span m="2432590">V2,</span> <span m="2434350">I</span> <span m="2434470">only</span> <span m="2434720">see</span> <span m="2435060">an edge</span> <span m="2435370">that</span> <span m="2435490">goes</span> <span m="2435720">from</span> <span m="2435840">V2</span> <span m="2436080">to</span> <span m="2436200">V3,</span> <span m="2437510">and</span> <span m="2437750">not</span> <span m="2437950">to</span> <span m="2438030">V2.</span> <span m="2438510">So</span> <span m="2439250">maybe</span> <span m="2439530">I</span> <span m="2439570">do</span> <span m="2439700">not</span> <span m="2439960">call</span> <span m="2440160">it</span> <span m="2440770">connected.</span> <span m="2441270">So</span> <span m="2441770">that's why</span> <span m="2442180">we</span> <span m="2442930">define</span> <span m="2443960">a</span> <span m="2444200">stronger</span> <span m="2444620">notion</span> <span m="2446480">for</span> <span m="2446730">a</span> <span m="2446810">digraph.</span> <span m="2448610">So</span> <span m="2448940">a</span> <span m="2449010">digraph,</span> <span m="2454180">G,</span> <span m="2455044">VE</span> <span m="2458170">is</span> <span m="2458410">called</span> <span m="2458700">strongly</span> <span m="2459230">connected</span> <span m="2469570">if</span> <span m="2470780">we</span> <span m="2471010">know</span> <span m="2471330">that</span> <span m="2472050">for</span> <span m="2472300">every</span> <span m="2472650">pair--</span> <span m="2475930">so</span> <span m="2476220">if</span> <span m="2476380">for</span> <span m="2476650">all</span> <span m="2477490">vertices</span> <span m="2478180">U</span> <span m="2478830">and</span> <span m="2479080">V</span> <span m="2480590">in</span> <span m="2480750">the</span> <span m="2480830">vertex</span> <span m="2481080">set--</span> <span m="2484270">there</span> <span m="2484440">exists</span> <span m="2485190">a</span> <span m="2485520">directed</span> <span m="2486330">path</span> <span m="2490630">that</span> <span m="2491880">starts</span> <span m="2492560">in</span> <span m="2493000">U</span> <span m="2494800">and</span> <span m="2495790">ends</span> <span m="2495950">up</span> <span m="2496310">in</span> <span m="2497300">V</span> <span m="2498120">in</span> <span m="2498530">G.</span> <span m="2499550">So</span> <span m="2499760">this</span> <span m="2500030">is</span> <span m="2500410">what</span> <span m="2500580">we</span> <span m="2500840">would</span> <span m="2501020">call</span> <span m="2501350">strongly</span> <span m="2501800">connected.</span> </p>
<p><span m="2506180">So</span> <span m="2506320">now</span> <span m="2506510">in</span> <span m="2506660">undirected</span> <span m="2507250">graphs,</span> <span m="2508450">we</span> <span m="2508590">had</span> <span m="2508930">connectivity</span> <span m="2509900">and</span> <span m="2510040">then</span> <span m="2510200">we</span> <span m="2510750">said,</span> <span m="2511010">well,</span> <span m="2511270">if a</span> <span m="2511540">connected,</span> <span m="2512410">undirected</span> <span m="2513010">graph</span> <span m="2513540">has</span> <span m="2513810">no</span> <span m="2514000">cycles.</span> <span m="2514740">We</span> <span m="2514990">have</span> <span m="2515230">trees</span> <span m="2515640">and they have</span> <span m="2516020">special</span> <span m="2516380">properties</span> <span m="2516910">and</span> <span m="2517030">all</span> <span m="2517160">that.</span> <span m="2517790">So</span> <span m="2517980">what</span> <span m="2518130">about</span> <span m="2518500">this</span> <span m="2518970">over</span> <span m="2519240">here?</span> <span m="2520370">Suppose</span> <span m="2520850">you</span> <span m="2520930">have</span> <span m="2521900">an</span> <span m="2522110">acyclic</span> <span m="2524480">strongly</span> <span m="2524950">connected</span> <span m="2525620">digraph.</span> <span m="2526820">What</span> <span m="2526930">does</span> <span m="2527100">that</span> <span m="2527280">look</span> <span m="2527440">like?</span> <span m="2529120">So</span> <span m="2529300">let's</span> <span m="2529520">give</span> <span m="2529640">an</span> <span m="2529730">example.</span> <span m="2532800">It's</span> <span m="2534490">not</span> <span m="2534610">completely</span> <span m="2535020">clear</span> <span m="2536770">what</span> <span m="2536870">kind</span> <span m="2537000">of</span> <span m="2537060">structure</span> <span m="2537330">it has.</span> <span m="2538070">So</span> <span m="2538330">for</span> <span m="2538480">example,</span> <span m="2544680">I</span> <span m="2544810">may</span> <span m="2545050">have</span> <span m="2545340">a</span> <span m="2545430">graph</span> <span m="2545820">that</span> <span m="2545940">looks</span> <span m="2546230">like</span> <span m="2546790">this,</span> <span m="2547130">for</span> <span m="2547170">example.</span> <span m="2554690">Say</span> <span m="2555690">this</span> <span m="2555910">is</span> <span m="2556010">an</span> <span m="2556190">acyclic</span> <span m="2557120">graph.</span> <span m="2559760">but</span> <span m="2566990">It</span> <span m="2567450">does</span> <span m="2567640">not</span> <span m="2567690">have</span> <span m="2567910">at all</span> <span m="2568390">a</span> <span m="2568480">tree</span> <span m="2568730">structure.</span> </p>
<p><span m="2570400">So</span> <span m="2571300">actually,</span> <span m="2571810">the</span> <span m="2571920">type</span> <span m="2572160">of</span> <span m="2572320">graph</span> <span m="2572970">the</span> <span m="2573110">we</span> <span m="2573280">have</span> <span m="2573570">here</span> <span m="2574500">is</span> <span m="2574730">called</span> <span m="2575230">a</span> <span m="2575390">directed</span> <span m="2575940">acyclic</span> <span m="2576510">graph.</span> <span m="2577940">As</span> <span m="2578030">you</span> <span m="2578130">can see,</span> <span m="2578510">there are no</span> <span m="2578690">cycles</span> <span m="2579330">because</span> <span m="2579980">I</span> <span m="2580190">only</span> <span m="2580530">go</span> <span m="2580710">forward,</span> <span m="2581350">essentially.</span> <span m="2581970">I can</span> <span m="2582030">never</span> <span m="2582380">go</span> <span m="2582580">backward</span> <span m="2583070">in</span> <span m="2583180">this</span> <span m="2583580">particular</span> <span m="2584890">way</span> <span m="2585180">that</span> <span m="2585390">I</span> <span m="2585450">depicted</span> <span m="2585960">the</span> <span m="2586000">graph.</span> <span m="2587370">So</span> <span m="2590000">this</span> <span m="2590250">is</span> <span m="2590460">an</span> <span m="2590530">example</span> <span m="2590970">for</span> <span m="2591090">directed</span> <span m="2591520">acyclic</span> <span m="2592040">graph.</span> <span m="2592930">But</span> <span m="2594050">it</span> <span m="2594160">doesn't</span> <span m="2594450">look</span> <span m="2594620">like</span> <span m="2594810">a</span> <span m="2594870">tree</span> <span m="2595100">at</span> <span m="2595180">all.</span> </p>
<p><span m="2596540">So</span> <span m="2596910">it's</span> <span m="2597140">worth</span> <span m="2597500">to</span> <span m="2597660">define</span> <span m="2598100">it</span> <span m="2598190">separately,</span> <span m="2598870">and</span> <span m="2598970">we</span> <span m="2599070">will</span> <span m="2599190">use</span> <span m="2599520">this</span> <span m="2602050">on</span> <span m="2602370">Thursday</span> <span m="2603010">when</span> <span m="2603460">we'll</span> <span m="2603600">talk</span> <span m="2603830">about</span> <span m="2604020">partial</span> <span m="2604350">orderings.</span> <span m="2605700">And</span> <span m="2606270">it</span> <span m="2606370">turns</span> <span m="2606730">out</span> <span m="2607450">that, as</span> <span m="2607730">you</span> <span m="2607830">can</span> <span m="2608000">see</span> <span m="2608330">here--</span> <span m="2609060">well,</span> <span m="2609780">at that</span> <span m="2610210">case,</span> <span m="2610460">a</span> <span m="2610770">directed</span> <span m="2611280">acyclic</span> <span m="2611680">graph</span> <span m="2611980">has</span> <span m="2612130">really</span> <span m="2612350">nice</span> <span m="2612640">properties,</span> <span m="2613230">and</span> <span m="2613330">one</span> <span m="2613480">of</span> <span m="2613590">them</span> <span m="2613790">is</span> <span m="2613960">that</span> <span m="2614070">you</span> <span m="2614160">can</span> <span m="2614330">order</span> <span m="2615540">these</span> <span m="2616090">vertices</span> <span m="2617165">in such a</span> <span m="2617450">way that</span> <span m="2617560">you</span> <span m="2617770">go</span> <span m="2618020">from,</span> <span m="2618780">say,</span> <span m="2619070">left</span> <span m="2619540">to</span> <span m="2619740">right</span> <span m="2620350">in</span> <span m="2620690">a</span> <span m="2620850">directed</span> <span m="2621260">fashion.</span> <span m="2623050">And</span> <span m="2623220">that</span> <span m="2623380">will</span> <span m="2623480">lead</span> <span m="2623750">to</span> <span m="2623860">partial</span> <span m="2624270">ordering.</span> <span m="2624930">So</span> <span m="2625260">that's</span> <span m="2625680">something</span> <span m="2626170">that</span> <span m="2626280">we</span> <span m="2626770">will</span> <span m="2627540">talk</span> <span m="2627830">about</span> <span m="2628080">as</span> <span m="2628200">well.</span> </p>
<p><span m="2629880">So</span> <span m="2630050">what's</span> <span m="2630180">the</span> <span m="2630220">definition?</span> <span m="2632200">A</span> <span m="2632840">directed</span> <span m="2635560">graph</span> <span m="2639030">is</span> <span m="2639370">called</span> <span m="2642280">a</span> <span m="2644760">directed</span> <span m="2647740">acyclic</span> <span m="2649420">graph,</span> <span m="2649890">and</span> <span m="2650110">we</span> <span m="2650780">appreciate</span> <span m="2651420">this</span> <span m="2651990">by</span> <span m="2652640">DAG.</span> <span m="2654250">We</span> <span m="2654380">call</span> <span m="2654590">them</span> <span m="2654850">DAGs.</span> <span m="2655690">They're</span> <span m="2655920">used</span> <span m="2657310">everywhere,</span> <span m="2657790">actually.</span> <span m="2666660">If it</span> <span m="2666700">does</span> <span m="2666890">not</span> <span m="2667070">contain</span> <span m="2669400">any</span> <span m="2671790">directed</span> <span m="2672990">cycles.</span> <span m="2678590">So</span> <span m="2678780">these</span> <span m="2679020">kinds</span> <span m="2679260">of</span> <span m="2679710">graphs are</span> <span m="2679790">used in</span> <span m="2680100">scheduling</span> <span m="2680800">and</span> <span m="2680960">optimization,</span> <span m="2683200">and</span> <span m="2683340">we will</span> <span m="2683590">use</span> <span m="2683860">them</span> <span m="2684330">next</span> <span m="2684630">week</span> <span m="2685050">in</span> <span m="2685870">the</span> <span m="2685980">lecture</span> <span m="2686430">on</span> <span m="2686610">partial</span> <span m="2687050">orderings.</span> <span m="2690680">Now</span> <span m="2690840">we</span> <span m="2690940">have</span> <span m="2691140">done</span> <span m="2691370">a</span> <span m="2691570">lot</span> <span m="2691730">of</span> <span m="2691840">definitions</span> <span m="2692510">concerning</span> <span m="2696000">directed</span> <span m="2696430">graphs.</span> </p>
<p><span m="2697330">So</span> <span m="2697800">now</span> <span m="2697980">let's</span> <span m="2698190">talk</span> <span m="2698390">about</span> <span m="2698640">these</span> <span m="2698850">very</span> <span m="2699030">special</span> <span m="2699430">ones,</span> <span m="2699680">tournament</span> <span m="2700080">graph,</span> <span m="2700420">and</span> <span m="2700540">see</span> <span m="2700710">whether we</span> <span m="2700940">can</span> <span m="2701080">prove</span> <span m="2701360">a</span> <span m="2701490">few</span> <span m="2701620">really</span> <span m="2701860">nice</span> <span m="2703230">theorems</span> <span m="2703640">about</span> <span m="2703940">them.</span> <span m="2708780">Let's</span> <span m="2708990">see</span> <span m="2709210">whether</span> <span m="2710000">we can</span> <span m="2710210">figure</span> <span m="2710710">that</span> <span m="2711060">out</span> <span m="2711550">together.</span> <span m="2722150">So</span> <span m="2722340">what</span> <span m="2722450">is</span> <span m="2722610">a</span> <span m="2722700">tournament</span> <span m="2723110">graph?</span> </p>
<p><span m="2736790">In</span> <span m="2736920">a</span> <span m="2737060">tournament</span> <span m="2737410">graph,</span> <span m="2737820">I</span> <span m="2737960">have</span> <span m="2738170">a</span> <span m="2738210">bunch</span> <span m="2738470">of</span> <span m="2738610">vertices.</span> <span m="2739760">And</span> <span m="2740820">essentially</span> <span m="2741560">we</span> <span m="2741750">want to</span> <span m="2742060">represent</span> <span m="2743380">like</span> <span m="2743590">a</span> <span m="2743640">tournament.</span> <span m="2743930">So</span> <span m="2744070">every</span> <span m="2744170">vertex,</span> <span m="2745330">say,</span> <span m="2745600">represents</span> <span m="2746700">a</span> <span m="2746850">team.</span> <span m="2747770">And</span> <span m="2747960">a</span> <span m="2748040">team</span> <span m="2748350">can</span> <span m="2749630">play</span> <span m="2750050">against</span> <span m="2750380">another</span> <span m="2750670">team,</span> <span m="2751070">and</span> <span m="2751250">beat</span> <span m="2751460">them</span> <span m="2751930">or</span> <span m="2752370">lose</span> <span m="2752910">against</span> <span m="2753630">the other</span> <span m="2753710">team.</span> <span m="2754660">So</span> <span m="2755800">we</span> <span m="2755950">want</span> <span m="2756120">to</span> <span m="2756390">use</span> <span m="2756800">the</span> <span m="2756860">directed</span> <span m="2757350">edges</span> <span m="2757700">to</span> <span m="2757860">indicate</span> <span m="2758670">who</span> <span m="2758850">is</span> <span m="2759000">winning</span> <span m="2759820">from</span> <span m="2760060">home.</span> <span m="2761050">And</span> <span m="2762400">such</span> <span m="2762700">types</span> <span m="2762980">of</span> <span m="2763090">graphs</span> <span m="2763450">have</span> <span m="2763640">very</span> <span m="2763810">special</span> <span m="2764170">property.</span> </p>
<p><span m="2765270">So</span> <span m="2765470">let</span> <span m="2765580">me</span> <span m="2765690">first</span> <span m="2767310">depict</span> <span m="2767690">one.</span> <span m="2768250">So</span> <span m="2768510">for</span> <span m="2768660">example,</span> <span m="2770320">we</span> <span m="2770500">have</span> <span m="2771920">E</span> <span m="2772680">goes</span> <span m="2773080">to</span> <span m="2773570">A,</span> <span m="2776400">goes</span> <span m="2776710">through</span> <span m="2776870">B,</span> <span m="2779060">incoming</span> <span m="2779500">edge</span> <span m="2779870">from</span> <span m="2780340">C,</span> <span m="2782610">one</span> <span m="2782890">coming</span> <span m="2783730">from</span> <span m="2784460">B.</span> <span m="2786700">And</span> <span m="2787600">over</span> <span m="2787910">here</span> <span m="2788130">we</span> <span m="2788210">have</span> <span m="2788450">this</span> <span m="2788540">directed</span> <span m="2788950">edge.</span> <span m="2793080">We</span> <span m="2793220">have</span> <span m="2793400">this</span> <span m="2793660">one.</span> <span m="2796160">We</span> <span m="2796540">have</span> <span m="2796760">this</span> <span m="2797020">one.</span> <span m="2797790">Let me</span> <span m="2797940">see</span> <span m="2798560">[INAUDIBLE]</span> <span m="2800130">don't</span> <span m="2800410">make</span> <span m="2800680">any</span> <span m="2800870">mistakes</span> <span m="2801460">here.</span> <span m="2804020">And</span> <span m="2804200">this</span> <span m="2804460">one.</span> <span m="2805790">And</span> <span m="2805990">over</span> <span m="2806250">here</span> <span m="2806400">is</span> <span m="2806550">another</span> <span m="2806750">one.</span> </p>
<p><span m="2809560">So</span> <span m="2809730">what</span> <span m="2809850">do</span> <span m="2809950">we</span> <span m="2810070">see</span> <span m="2810290">in</span> <span m="2810400">this</span> <span m="2810590">graph?</span> <span m="2811470">We</span> <span m="2811500">see</span> <span m="2811900">that</span> <span m="2812900">either</span> <span m="2817040">say,</span> <span m="2817440">team</span> <span m="2817800">U,</span> <span m="2819140">beats</span> <span m="2821460">team</span> <span m="2821950">V.</span> <span m="2822840">And</span> <span m="2823050">that</span> <span m="2823260">means</span> <span m="2823590">that</span> <span m="2823830">we</span> <span m="2823920">have</span> <span m="2824090">a</span> <span m="2824160">directed</span> <span m="2824710">edge</span> <span m="2825030">from</span> <span m="2825250">U</span> <span m="2826080">pointing</span> <span m="2826600">at</span> <span m="2826860">V.</span> <span m="2828190">Or</span> <span m="2830340">it's</span> <span m="2830500">other </span> <span m="2830700">way</span> <span m="2830830">around.</span> <span m="2832200">V</span> <span m="2832610">actually</span> <span m="2833030">beats</span> <span m="2833550">U,</span> <span m="2834710">and</span> <span m="2834880">we</span> <span m="2834970">have</span> <span m="2835130">a</span> <span m="2835280">directed</span> <span m="2835600">edge</span> <span m="2835860">from</span> <span m="2836070">V</span> <span m="2836775">to</span> <span m="2837090">U.</span> </p>
<p><span m="2839970">Let's</span> <span m="2840160">have</span> <span m="2840300">a</span> <span m="2840370">look</span> <span m="2840560">at</span> <span m="2840600">this</span> <span m="2840790">graph</span> <span m="2841620">and</span> <span m="2841810">see</span> <span m="2844390">how</span> <span m="2844560">this</span> <span m="2844740">works.</span> <span m="2845140">So</span> <span m="2846280">for</span> <span m="2846420">example,</span> <span m="2847500">we</span> <span m="2847550">have</span> <span m="2847920">that</span> <span m="2848490">B</span> <span m="2849300">is</span> <span m="2849520">beating</span> <span m="2850000">E,</span> <span m="2850360">and</span> <span m="2850570">E</span> <span m="2850800">is beating</span> <span m="2851060">A,</span> <span m="2851320">and</span> <span m="2851970">so</span> <span m="2852190">on.</span> <span m="2853150">So</span> <span m="2853220">let's</span> <span m="2853410">have</span> <span m="2853570">a</span> <span m="2853640">look.</span> <span m="2854030">Maybe</span> <span m="2854240">we</span> <span m="2854340">can</span> <span m="2855140">figure</span> <span m="2855480">out</span> <span m="2855670">who's</span> <span m="2855900">the</span> <span m="2855970">best</span> <span m="2856260">player</span> <span m="2856640">in</span> <span m="2856770">here.</span> </p>
<p><span m="2859540">So</span> <span m="2860290">this</span> <span m="2860450">is</span> <span m="2860530">sort</span> <span m="2860780">of</span> <span m="2860880">a</span> <span m="2860950">general</span> <span m="2861300">question</span> <span m="2861660">if</span> <span m="2861870">who</span> <span m="2861880">would</span> <span m="2861990">like</span> <span m="2862160">to</span> <span m="2862320">answer.</span> <span m="2862580">Maybe you</span> <span m="2862960">cannot</span> <span m="2863340">answer</span> <span m="2863660">this.</span> <span m="2864690">So</span> <span m="2864910">let's</span> <span m="2865090">have</span> <span m="2865230">an</span> <span m="2865320">example.</span> <span m="2866910">Has</span> <span m="2867255">anybody</span> <span m="2867750">seen</span> <span m="2868290">an</span> <span m="2868350">example</span> <span m="2869230">where</span> <span m="2873570">we</span> <span m="2873750">start</span> <span m="2874220">with</span> <span m="2874570">A,</span> <span m="2875980">then</span> <span m="2876700">we</span> <span m="2877500">may</span> <span m="2877780">beat</span> <span m="2878060">another</span> <span m="2878360">vertex,</span> <span m="2878690">and</span> <span m="2879180">maybe</span> <span m="2879490">another</span> <span m="2879690">vertex,</span> <span m="2880350">and</span> <span m="2880540">so</span> <span m="2880770">on,</span> <span m="2881510">until</span> <span m="2881890">we</span> <span m="2882010">have</span> <span m="2882160">covered</span> <span m="2882460">all</span> <span m="2882620">the</span> <span m="2882710">different</span> <span m="2883030">vertices.</span> </p>
<p><span m="2884540">Do</span> <span m="2884610">you</span> <span m="2884730">see</span> <span m="2884950">a</span> <span m="2885020">path</span> <span m="2885450">that</span> <span m="2886130">works</span> <span m="2886450">like</span> <span m="2886670">that?</span> <span m="2887530">And</span> <span m="2888060">that could</span> <span m="2888390">gives</span> <span m="2888640">us</span> <span m="2888860">an</span> <span m="2889090">ordering</span> <span m="2890110">on</span> <span m="2890570">who</span> <span m="2890760">is</span> <span m="2890900">the</span> <span m="2890970">best</span> <span m="2891230">player,</span> <span m="2891930">like</span> <span m="2892160">the</span> <span m="2892290">one</span> <span m="2894020">at</span> <span m="2894220">the</span> <span m="2894360">top,</span> <span m="2894840">like</span> <span m="2895140">A,</span> <span m="2895520">is</span> <span m="2895900">able</span> <span m="2896200">to,</span> <span m="2896650">for</span> <span m="2896720">example,</span> <span m="2897260">beat</span> <span m="2899020">B.</span> <span m="2900030">And</span> <span m="2900240">B</span> <span m="2900430">is,</span> <span m="2900560">for</span> <span m="2900670">example,</span> <span m="2901350">able</span> <span m="2901690">to</span> <span m="2901830">beat</span> <span m="2903310">D</span> <span m="2904210">and</span> <span m="2904650">this</span> <span m="2904880">one,</span> <span m="2905880">E,</span> <span m="2907170">and</span> <span m="2907390">this</span> <span m="2907620">one,</span> <span m="2907970">C.</span> <span m="2908520">So</span> <span m="2909050">you</span> <span m="2909220">would</span> <span m="2909350">say,</span> <span m="2909680">OK,</span> <span m="2909950">that's</span> <span m="2910200">great.</span> <span m="2910660">Now</span> <span m="2910730">I</span> <span m="2910810">know</span> <span m="2910980">that</span> <span m="2911080">this</span> <span m="2911240">one</span> <span m="2911390">is</span> <span m="2911510">the</span> <span m="2911590">strongest</span> <span m="2912070">player.</span> </p>
<p><span m="2913850">But</span> <span m="2914160">there's</span> <span m="2914620">a</span> <span m="2914670">little</span> <span m="2914880">problem</span> <span m="2915250">here,</span> <span m="2915490">right?</span> <span m="2916810">Because</span> <span m="2917775">I</span> <span m="2918170">can</span> <span m="2918370">produce</span> <span m="2918720">many</span> <span m="2919000">such</span> <span m="2919290">paths.</span> <span m="2919600">And</span> <span m="2919700">actually,</span> <span m="2920180">if</span> <span m="2920360">I</span> <span m="2920430">look</span> <span m="2920720">at</span> <span m="2920910">C,</span> <span m="2922140">then</span> <span m="2922330">C</span> <span m="2924210">beats</span> <span m="2924410">A</span> <span m="2924810">as well.</span> <span m="2925060">So</span> <span m="2925190">that's</span> <span m="2925350">kind</span> <span m="2925550">of</span> <span m="2925620">weird.</span> <span m="2927260">So</span> <span m="2928070">wait</span> <span m="2928260">a</span> <span m="2928290">minute.</span> <span m="2931820">We</span> <span m="2931940">have</span> <span m="2932180">that</span> <span m="2932330">there's</span> <span m="2932480">a</span> <span m="2932810">directed</span> <span m="2933130">edge</span> <span m="2933350">from</span> <span m="2933570">C</span> <span m="2934000">to</span> <span m="2934060">A.</span> <span m="2936950">It's</span> <span m="2937180">like</span> <span m="2937860">teams</span> <span m="2938310">beats</span> <span m="2938570">one</span> <span m="2938760">another.</span> <span m="2939120">And</span> <span m="2939330">it's not</span> <span m="2939660">very</span> <span m="2939860">clear</span> <span m="2940140">how we</span> <span m="2940390">can</span> <span m="2940530">talk</span> <span m="2940760">about</span> <span m="2942710">a</span> <span m="2942840">best</span> <span m="2943560">player.</span> </p>
<p><span m="2945800">Well,</span> <span m="2945940">we</span> <span m="2946050">would</span> <span m="2946240">have</span> <span m="2946400">a</span> <span m="2946470">best</span> <span m="2946770">player</span> <span m="2947180">if</span> <span m="2947950">one</span> <span m="2948620">player</span> <span m="2949120">sort</span> <span m="2949360">of</span> <span m="2952390">wins</span> <span m="2952760">from</span> <span m="2952980">everybody</span> <span m="2953500">else.</span> <span m="2955090">But</span> <span m="2955220">there's</span> <span m="2955490">many</span> <span m="2955730">examples</span> <span m="2956290">here.</span> <span m="2956820">So</span> <span m="2957610">let's</span> <span m="2958070">look</span> <span m="2958270">at</span> <span m="2958330">another</span> <span m="2959010">walk.</span> <span m="2959840">For</span> <span m="2959990">example,</span> <span m="2960420">C</span> <span m="2961230">can</span> <span m="2961400">go</span> <span m="2961570">to</span> <span m="2961770">B,</span> <span m="2963010">to</span> <span m="2963160">D,</span> <span m="2963455">and</span> <span m="2963750">then</span> <span m="2963940">to</span> <span m="2964180">E,</span> <span m="2965140">and</span> <span m="2965360">then</span> <span m="2965550">to</span> <span m="2965790">A.</span> <span m="2966140">So there are</span> <span m="2966380">many</span> <span m="2966690">possibilities</span> <span m="2967380">here.</span> </p>
<p><span m="2968280">So</span> <span m="2968420">this</span> <span m="2968620">leads us</span> <span m="2969090">to</span> <span m="2969450">a</span> <span m="2970260">concept.</span> <span m="2973640">And we</span> <span m="2973960">call</span> <span m="2974230">that's</span> <span m="2974440">a</span> <span m="2978070">directed</span> <span m="2978510">Hamiltonian</span> <span m="2979010">path.</span> <span m="2980140">And</span> <span m="2980290">we're</span> <span m="2980380">going</span> <span m="2980590">to</span> <span m="2980710">show</span> <span m="2981580">that,</span> <span m="2981790">in</span> <span m="2981980">a</span> <span m="2982060">tournament</span> <span m="2982480">graph,</span> <span m="2983260">you</span> <span m="2983370">can</span> <span m="2983570">always</span> <span m="2984040">find</span> <span m="2984300">such</span> <span m="2984580">a</span> <span m="2984630">directed</span> <span m="2985270">Hamiltonian</span> <span m="2985760">path.</span> </p>
<p><span m="2986120">So what's a</span> <span m="2986570">Hamiltonian</span> <span m="2986920">path?</span> <span m="2988060">This</span> <span m="2988220">is</span> <span m="2988340">actually</span> <span m="2988670">an example</span> <span m="2989190">of</span> <span m="2989360">it.</span> <span m="2990360">There's</span> <span m="2990730">a</span> <span m="2990820">walk</span> <span m="2991560">that</span> <span m="2991750">goes</span> <span m="2992130">around</span> <span m="2992690">the</span> <span m="2992780">graph</span> <span m="2993710">and</span> <span m="2993920">visits</span> <span m="2994430">every</span> <span m="2995020">vertex</span> <span m="2995900">exactly</span> <span m="2996410">once.</span> <span m="2997240">So</span> <span m="2997390">we're</span> <span m="2997950">going</span> <span m="2998110">to</span> <span m="2998250">prove</span> <span m="2998690">that</span> <span m="2999780">a</span> <span m="2999910">tournament</span> <span m="3000320">graph</span> <span m="3000640">has</span> <span m="3000870">this</span> <span m="3001280">beautiful</span> <span m="3001720">property.</span> </p>
<p><span m="3018070">So</span> <span m="3018300">let's</span> <span m="3018710">first</span> <span m="3019570">write</span> <span m="3019840">out</span> <span m="3019960">a</span> <span m="3020050">definition</span> <span m="3020720">of</span> <span m="3020810">this.</span> <span m="3026620">A</span> <span m="3026720">directed</span> <span m="3030910">Hamiltonian</span> <span m="3033960">path</span> <span m="3040610">is</span> <span m="3041020">a</span> <span m="3041460">directed</span> <span m="3042330">walk</span> <span m="3048140">that</span> <span m="3048400">visits</span> <span m="3055870">every</span> <span m="3057210">vertex</span> <span m="3062520">exactly</span> <span m="3065120">once.</span> <span m="3068090">So</span> <span m="3068320">as</span> <span m="3068480">I</span> <span m="3068520">said</span> <span m="3068840">already,</span> <span m="3069230">here</span> <span m="3069550">we</span> <span m="3069680">have</span> <span m="3069870">such</span> <span m="3070140">an</span> <span m="3070190">example.</span> <span m="3072630">We</span> <span m="3072640">can</span> <span m="3072800">go</span> <span m="3072990">from</span> <span m="3073240">A</span> <span m="3073470">to</span> <span m="3073680">B,</span> <span m="3074020">to</span> <span m="3074280">D</span> <span m="3074540">to E,</span> <span m="3074750">to</span> <span m="3074900">C.</span> <span m="3075890">Maybe</span> <span m="3076140">there</span> <span m="3076350">are</span> <span m="3076570">even</span> <span m="3076670">other</span> <span m="3077160">examples.</span> <span m="3078440">I</span> <span m="3078540">did</span> <span m="3078680">not</span> <span m="3078820">actually</span> <span m="3079640">see</span> <span m="3079930">them.</span> <span m="3080160">Maybe</span> <span m="3080380">you</span> <span m="3080450">can</span> <span m="3080590">have</span> <span m="3080800">a</span> <span m="3080850">look</span> <span m="3081040">at</span> <span m="3081090">them</span> <span m="3081270">as</span> <span m="3081400">well.</span> <span m="3082540">So</span> <span m="3082720">maybe</span> <span m="3082900">there's</span> <span m="3083070">something</span> <span m="3083410">that</span> <span m="3083490">starts</span> <span m="3083820">with</span> <span m="3084160">B</span> <span m="3084550">going</span> <span m="3084870">to</span> <span m="3085170">E</span> <span m="3085380">maybe.</span> <span m="3085800">That's</span> <span m="3085970">a</span> <span m="3086060">very</span> <span m="3086250">different</span> <span m="3087210">direction,</span> <span m="3088580">like</span> <span m="3088780">this.</span> <span m="3090660">There</span> <span m="3090730">will</span> <span m="3090850">be</span> <span m="3090990">one</span> <span m="3091200">as</span> <span m="3091330">well,</span> <span m="3092920">and</span> <span m="3093120">so</span> <span m="3093250">forth.</span> </p>
<p><span m="3095960">So</span> <span m="3096230">what's</span> <span m="3096720">the</span> <span m="3096890">theorem</span> <span m="3097250">that</span> <span m="3097620">you</span> <span m="3097730">want</span> <span m="3097880">to</span> <span m="3097960">prove?</span> <span m="3098960">The</span> <span m="3099070">theorem</span> <span m="3099510">in</span> <span m="3100560">that</span> <span m="3100990">you</span> <span m="3101110">want</span> <span m="3101230">to</span> <span m="3101310">show</span> <span m="3101600">that</span> <span m="3101720">every</span> <span m="3106390">tournament</span> <span m="3109160">graph</span> <span m="3112490">actually</span> <span m="3112950">contains</span> <span m="3118320">such</span> <span m="3118860">a</span> <span m="3118930">directed</span> <span m="3119340">Hamiltonian</span> <span m="3119960">path.</span> <span m="3136710">So</span> <span m="3137020">let's</span> <span m="3137640">have</span> <span m="3137960">a</span> <span m="3138040">look</span> <span m="3138220">at</span> <span m="3138290">how</span> <span m="3138430">we</span> <span m="3138520">can</span> <span m="3138630">prove</span> <span m="3138990">this.</span> </p>
<p><span m="3158960">What</span> <span m="3159080">kind</span> <span m="3159220">of</span> <span m="3159310">proof</span> <span m="3159530">technique</span> <span m="3159910">are we</span> <span m="3160110">going</span> <span m="3160360">to</span> <span m="3160490">use?</span> <span m="3165220">Do</span> <span m="3165540">you</span> <span m="3165780">have</span> <span m="3165960">an idea?</span> <span m="3169010">Well,</span> <span m="3169160">usually we</span> <span m="3169330">use</span> <span m="3169670">induction</span> <span m="3170220">so</span> <span m="3170280">that's what</span> <span m="3170470">we're</span> <span m="3170690">going</span> <span m="3170900">to</span> <span m="3170980">do</span> <span m="3171150">here</span> <span m="3171430">as</span> <span m="3171560">well.</span> <span m="3172310">But what</span> <span m="3172490">kind</span> <span m="3172710">of</span> <span m="3173360">induction</span> <span m="3173850">hypothesis</span> <span m="3174380">can we</span> <span m="3174660">do?</span> <span m="3175770">So</span> <span m="3175960">what</span> <span m="3176270">would we</span> <span m="3176680">induct</span> <span m="3177200">on,</span> <span m="3177610">you think?</span> <span m="3180834">Someone</span> <span m="3181308">else?</span> <span m="3181782">Maybe someone</span> <span m="3182256">up there?</span> </p>
<p><span m="3184626">AUDIENCE: The</span> <span m="3185100">number of</span> <span m="3185574">nodes.</span> </p>
<p><span m="3186060">PROFESSOR: The number of</span> <span m="3187075">nodes.</span> <span m="3188670">So</span> <span m="3191830">we</span> <span m="3191950">use</span> <span m="3192150">induction</span> <span m="3193100">on</span> <span m="3193480">the</span> <span m="3193590">number</span> <span m="3193870">of</span> <span m="3193980">nodes.</span> <span m="3196050">And</span> <span m="3196200">why</span> <span m="3196350">would</span> <span m="3196520">that</span> <span m="3196680">be</span> <span m="3196860">of</span> <span m="3196970">interest?</span> <span m="3197930">So</span> <span m="3198730">let's</span> <span m="3198900">have a</span> <span m="3199110">look</span> <span m="3199820">at</span> <span m="3199870">how</span> <span m="3200060">we</span> <span m="3200160">can</span> <span m="3200300">think of</span> <span m="3200640">that.</span> <span m="3201130">So</span> <span m="3201300">we</span> <span m="3201490">start</span> <span m="3201770">thinking</span> <span m="3202050">about</span> <span m="3202240">such</span> <span m="3202450">a</span> <span m="3202510">problem,</span> <span m="3203600">this</span> <span m="3203850">really</span> <span m="3204500">sort</span> <span m="3204790">of--</span> <span m="3206920">one</span> <span m="3207130">parameter</span> <span m="3207750">here</span> <span m="3207980">that's</span> <span m="3208180">the</span> <span m="3208210">number</span> <span m="3208450">of</span> <span m="3208570">nodes</span> <span m="3209070">in</span> <span m="3210450">a</span> <span m="3210530">tournament</span> <span m="3210990">graph.</span> <span m="3211960">I</span> <span m="3212130">also</span> <span m="3212330">have</span> <span m="3212560">edges.</span> <span m="3213580">But</span> <span m="3214260">if</span> <span m="3214820">I</span> <span m="3214880">think</span> <span m="3215130">about</span> <span m="3215560">edges,</span> <span m="3216460">then,</span> <span m="3220395">the</span> <span m="3220770">edges</span> <span m="3221290">is</span> <span m="3221480">always</span> <span m="3224010">directly</span> <span m="3224410">related</span> <span m="3224810">to</span> <span m="3225020">the</span> <span m="3225110">number</span> <span m="3225340">of</span> <span m="3225480">nodes</span> <span m="3225930">in</span> <span m="3226390">a</span> <span m="3226470">tournament</span> <span m="3226840">graph.</span> <span m="3227430">So</span> <span m="3227560">I</span> <span m="3227640">really</span> <span m="3228410">have</span> <span m="3228620">just</span> <span m="3228910">that</span> <span m="3229200">one</span> <span m="3229380">parameter.</span> </p>
<p><span m="3230640">So</span> <span m="3230930">it</span> <span m="3231420">makes</span> <span m="3231620">sense</span> <span m="3231890">to</span> <span m="3231970">use</span> <span m="3232410">induction</span> <span m="3232900">on</span> <span m="3233100">number</span> <span m="3233320">of</span> <span m="3233430">nodes.</span> <span m="3235040">So</span> <span m="3237400">induct</span> <span m="3237710">on</span> <span m="3237860">n,</span> <span m="3238440">where</span> <span m="3239440">Pn</span> <span m="3240940">is</span> <span m="3241240">going</span> <span m="3241460">to</span> <span m="3241580">be</span> <span m="3241830">that</span> <span m="3242200">every--</span> <span m="3244080">and</span> <span m="3244220">essentially,</span> <span m="3244590">the</span> <span m="3244670">theorem</span> <span m="3245070">holds</span> <span m="3245400">true</span> <span m="3245650">for</span> <span m="3245830">a</span> <span m="3245890">tournament</span> <span m="3246280">graph</span> <span m="3246730">on</span> <span m="3247080">n</span> <span m="3247290">nodes--</span> <span m="3248230">so</span> <span m="3248540">every</span> <span m="3250360">tournament</span> <span m="3250600">graph</span> <span m="3256520">on</span> <span m="3257300">n</span> <span m="3257640">nodes</span> <span m="3259800">actually</span> <span m="3260190">contains</span> <span m="3268670">a</span> <span m="3268800">directed</span> <span m="3269260">Hamiltonian</span> <span m="3269890">path.</span> </p>
<p><span m="3272820">So</span> <span m="3272970">this</span> <span m="3273190">is</span> <span m="3273320">semi</span> <span m="3273700">induction</span> <span m="3274090">hypothesis.</span> <span m="3276440">So</span> <span m="3276480">when</span> <span m="3276650">you</span> <span m="3277010">think</span> <span m="3277260">about</span> <span m="3277600">that,</span> <span m="3278320">well,</span> <span m="3278940">I</span> <span m="3279050">feel</span> <span m="3279250">pretty</span> <span m="3279490">confident</span> <span m="3279920">because</span> <span m="3280170">if</span> <span m="3280330">I</span> <span m="3280370">look</span> <span m="3280590">at</span> <span m="3280650">the</span> <span m="3280730">base</span> <span m="3281110">case--</span> <span m="3281500">that's</span> <span m="3281640">how</span> <span m="3281770">we</span> <span m="3281900">always</span> <span m="3282240">start--</span> <span m="3284620">well,</span> <span m="3285430">n</span> <span m="3285690">equals</span> <span m="3286130">1.</span> <span m="3286730">If</span> <span m="3286900">n</span> <span m="3287120">equals</span> <span m="3287430">1,</span> <span m="3287700">I</span> <span m="3287900">have</span> <span m="3288000">just</span> <span m="3288230">a</span> <span m="3288270">single</span> <span m="3288640">vertex.</span> <span m="3289440">There's</span> <span m="3289660">no</span> <span m="3289960">edges.</span> <span m="3290420">Everything</span> <span m="3290800">is</span> <span m="3290950">fine</span> <span m="3291270">because</span> <span m="3292150">the</span> <span m="3292270">single</span> <span m="3292730">edge</span> <span m="3293130">is</span> <span m="3293460">a</span> <span m="3293630">directed</span> <span m="3294150">Hamiltonian</span> <span m="3294590">path.</span> </p>
<p><span m="3295660">So</span> <span m="3296580">this</span> <span m="3296780">is</span> <span m="3296920">great.</span> <span m="3297280">So</span> <span m="3297460">this</span> <span m="3297650">works.</span> <span m="3299250">So</span> <span m="3299350">what</span> <span m="3299490">about</span> <span m="3299760">inductive</span> <span m="3300200">step?</span> <span m="3303460">Now</span> <span m="3303610">with</span> <span m="3303760">inductive</span> <span m="3304220">step,</span> <span m="3305090">we</span> <span m="3305260">always</span> <span m="3305700">perceived</span> <span m="3306250">in</span> <span m="3306370">the</span> <span m="3306450">same</span> <span m="3306770">way.</span> <span m="3307800">We</span> <span m="3307900">start</span> <span m="3308210">to</span> <span m="3308320">assume</span> <span m="3311450">that</span> <span m="3312900">Pn</span> <span m="3313220">is true.</span> </p>
<p><span m="3313540">Essentially,</span> <span m="3313970">the</span> <span m="3314080">theorem</span> <span m="3314500">holds</span> <span m="3314860">for</span> <span m="3315060">a</span> <span m="3315220">tournament</span> <span m="3315390">graph</span> <span m="3316000">on</span> <span m="3316530">n</span> <span m="3317027">vertices.</span> <span m="3320510">Actually,</span> <span m="3321325">let</span> <span m="3321740">me</span> <span m="3322215">keep</span> <span m="3322690">this</span> <span m="3322810">over here.</span> <span m="3329570">So</span> <span m="3329790">now</span> <span m="3329960">let's</span> <span m="3330260">just</span> <span m="3330500">think</span> <span m="3330770">about</span> <span m="3331660">how</span> <span m="3331960">we</span> <span m="3332100">can</span> <span m="3332250">prove</span> <span m="3332590">this</span> <span m="3332770">induction</span> <span m="3333150">step.</span> <span m="3335090">I</span> <span m="3335180">need</span> <span m="3335420">to</span> <span m="3335530">prove</span> <span m="3335960">P</span> <span m="3336910">of</span> <span m="3337140">n plus</span> <span m="3337530">1.</span> <span m="3338570">So</span> <span m="3339190">how</span> <span m="3339340">do</span> <span m="3339460">I</span> <span m="3339550">start?</span> <span m="3340050">I</span> <span m="3340120">have</span> <span m="3340310">to</span> <span m="3340430">start</span> <span m="3340960">with</span> <span m="3341450">a</span> <span m="3341570">tournament</span> <span m="3342000">graph</span> <span m="3342640">on</span> <span m="3342860">n</span> <span m="3343060">plus</span> <span m="3343300">1</span> <span m="3343470">vertices.</span> <span m="3344950">And</span> <span m="3345190">somehow</span> <span m="3346840">I</span> <span m="3346940">got</span> <span m="3347110">to</span> <span m="3347190">be</span> <span m="3347360">able</span> <span m="3347510">to</span> <span m="3347610">use</span> <span m="3348090">this</span> <span m="3349910">property</span> <span m="3350360">because</span> <span m="3350610">that's</span> <span m="3350780">what</span> <span m="3350930">I</span> <span m="3351050">assume.</span> <span m="3353060">And</span> <span m="3353180">the</span> <span m="3353280">property</span> <span m="3353700">only</span> <span m="3353930">holds</span> <span m="3354290">for</span> <span m="3354720">a</span> <span m="3354830">tournament</span> <span m="3355150">graph</span> <span m="3355410">on</span> <span m="3355570">end</span> <span m="3355790">points.</span> <span m="3356330">So</span> <span m="3356530">what</span> <span m="3356680">will</span> <span m="3356770">be a</span> <span m="3356970">really</span> <span m="3357200">good</span> <span m="3357350">strategy</span> <span m="3357930">to</span> <span m="3358080">do</span> <span m="3358310">here</span> <span m="3358900">to</span> <span m="3359060">sort</span> <span m="3359260">of</span> <span m="3359340">proceed</span> <span m="3359790">our</span> <span m="3359980">proof?</span> </p>
<p><span m="3360900">So</span> <span m="3361100">let</span> <span m="3361270">me</span> <span m="3362390">first</span> <span m="3363160">write</span> <span m="3363470">out</span> <span m="3364400">what</span> <span m="3364720">we</span> <span m="3365550">want</span> <span m="3365730">to</span> <span m="3365850">do</span> <span m="3366100">here.</span> <span m="3368380">Maybe</span> <span m="3368590">you</span> <span m="3368680">can</span> <span m="3368820">think</span> <span m="3369110">about</span> <span m="3370190">how</span> <span m="3370350">to</span> <span m="3371900">advance</span> <span m="3372460">here.</span> <span m="3375040">So</span> <span m="3375660">we</span> <span m="3375810">have</span> <span m="3376210">shown</span> <span m="3376610">Pn.</span> <span m="3378460">Now</span> <span m="3379200">we</span> <span m="3379340">want</span> <span m="3379580">to</span> <span m="3381100">actually</span> <span m="3381830">prove</span> <span m="3382100">something</span> <span m="3382840">about</span> <span m="3383940">tournament</span> <span m="3384290">graphs</span> <span m="3384600">on</span> <span m="3384720">n</span> <span m="3384850">plus</span> <span m="3385070">1</span> <span m="3385530">nodes, so</span> <span m="3386000">let's</span> <span m="3386220">consider</span> <span m="3386670">one.</span> </p>
<p><span m="3388250">Consider</span> <span m="3388960">a</span> <span m="3389100">tournament</span> <span m="3392350">graph</span> <span m="3395760">on</span> <span m="3396510">n</span> <span m="3396880">plus</span> <span m="3397210">1</span> <span m="3398390">nodes.</span> <span m="3401310">Now</span> <span m="3401520">how</span> <span m="3402060">can</span> <span m="3402310">I</span> <span m="3402590">use</span> <span m="3403010">my</span> <span m="3406270">induction</span> <span m="3406640">hypothesis?</span> <span m="3407970">So</span> <span m="3409390">I</span> <span m="3409520">start</span> <span m="3409870">with</span> <span m="3410070">this,</span> <span m="3410630">and</span> <span m="3410810">I</span> <span m="3410840">want</span> <span m="3411070">to</span> <span m="3411140">use</span> <span m="3412390">something</span> <span m="3412850">that</span> <span m="3413010">talks</span> <span m="3413290">about</span> <span m="3413650">the</span> <span m="3413750">tournament</span> <span m="3414020">graph</span> <span m="3414360">on</span> <span m="3414520">n</span> <span m="3414730">nodes.</span> <span m="3415780">So</span> <span m="3415960">how</span> <span m="3416190">could I</span> <span m="3416460">proceed</span> <span m="3416850">here?</span> <span m="3422616">Any</span> <span m="3423102">suggestions?</span> <span m="3424560">So</span> <span m="3424660">what</span> <span m="3424945">do you</span> <span m="3425490">usually</span> <span m="3425890">do,</span> <span m="3426710">right?</span> <span m="3427130">If I</span> <span m="3427550">have</span> <span m="3427970">like</span> <span m="3428420">an</span> <span m="3428870">n</span> <span m="3428910">plus</span> <span m="3429060">1</span> <span m="3429330">nodes,</span> <span m="3429850">I</span> <span m="3430275">have to</span> <span m="3430700">somehow</span> <span m="3431140">look</span> <span m="3431620">at</span> <span m="3432120">least</span> <span m="3432440">maybe</span> <span m="3432680">there</span> <span m="3433000">exists</span> <span m="3433290">a</span> <span m="3433450">subgraph</span> <span m="3434065">in this</span> <span m="3434360">bigger</span> <span m="3434480">graph.</span> <span m="3435240">So</span> <span m="3435390">this</span> <span m="3435590">is</span> <span m="3435780">really</span> <span m="3435960">how</span> <span m="3436130">you</span> <span m="3436660">always</span> <span m="3437120">think</span> <span m="3437330">about</span> <span m="3437580">these</span> <span m="3437660">types of</span> <span m="3437960">proofs</span> <span m="3438530">or also</span> <span m="3438870">other</span> <span m="3438940">problems.</span> </p>
<p><span m="3441140">So</span> <span m="3441290">there</span> <span m="3441380">must</span> <span m="3441600">be</span> <span m="3441730">some</span> <span m="3441920">kind</span> <span m="3442110">of</span> <span m="3442190">subgraph</span> <span m="3443340">that</span> <span m="3443550">already</span> <span m="3444030">has</span> <span m="3444410">this</span> <span m="3444990">property.</span> <span m="3446400">Well,</span> <span m="3449350">let's</span> <span m="3449980">take</span> <span m="3450250">out</span> <span m="3450490">one</span> <span m="3450760">node</span> <span m="3451160">and</span> <span m="3451270">see</span> <span m="3451440">what</span> <span m="3451590">happens</span> <span m="3453420">because</span> <span m="3453900">then</span> <span m="3453990">we</span> <span m="3454330">have</span> <span m="3454610">one</span> <span m="3454820">node</span> <span m="3455050">less</span> <span m="3455470">and</span> <span m="3455630">maybe</span> <span m="3456670">we</span> <span m="3456840">will</span> <span m="3456950">be</span> <span m="3457110">able</span> <span m="3457250">to</span> <span m="3457340">apply</span> <span m="3458120">our</span> <span m="3459320">induction</span> <span m="3459730">step.</span> <span m="3461380">So</span> <span m="3461550">let's</span> <span m="3462660">take</span> <span m="3462970">out</span> <span m="3465820">one</span> <span m="3466140">node</span> <span m="3466660">V.</span> </p>
<p><span m="3471550">And</span> <span m="3472450">what can we</span> <span m="3472620">say</span> <span m="3473060">about</span> <span m="3473280">remaining</span> <span m="3473660">graph</span> <span m="3474740">if</span> <span m="3474950">we</span> <span m="3475190">take</span> <span m="3475430">out</span> <span m="3475620">one</span> <span m="3475810">node?</span> <span m="3476040">For</span> <span m="3476170">example,</span> <span m="3476580">if</span> <span m="3476640">I</span> <span m="3476690">take</span> <span m="3476990">out</span> <span m="3477870">the</span> <span m="3477970">node</span> <span m="3478270">E</span> <span m="3478520">over</span> <span m="3478790">here</span> <span m="3479670">and</span> <span m="3479820">I</span> <span m="3479870">just</span> <span m="3480110">look</span> <span m="3480260">at</span> <span m="3480330">all</span> <span m="3480450">the</span> <span m="3480510">rest,</span> <span m="3481280">I</span> <span m="3481400">can</span> <span m="3481580">still</span> <span m="3481940">see</span> <span m="3482300">that</span> <span m="3482880">for</span> <span m="3483100">all</span> <span m="3483270">the</span> <span m="3483410">other</span> <span m="3483640">nodes</span> <span m="3485180">either,</span> <span m="3485900">say,</span> <span m="3486220">U</span> <span m="3486520">beats</span> <span m="3486965">V</span> <span m="3487410">or</span> <span m="3487710">V</span> <span m="3487940">beats</span> <span m="3488255">U.</span> <span m="3488570">So</span> <span m="3488790">I</span> <span m="3488860">still</span> <span m="3489100">have</span> <span m="3489980">an</span> <span m="3490250">edge</span> <span m="3490890">in</span> <span m="3491140">one</span> <span m="3491330">direction</span> <span m="3492600">between</span> <span m="3493160">each</span> <span m="3493480">two</span> <span m="3494180">nodes.</span> </p>
<p><span m="3495130">So</span> <span m="3495260">actually</span> <span m="3495560">I</span> <span m="3495600">still</span> <span m="3495900">have</span> <span m="3496070">a</span> <span m="3496150">tournament</span> <span m="3496550">graph,</span> <span m="3497350">so</span> <span m="3497500">that's</span> <span m="3497720">great.</span> <span m="3503340">So</span> <span m="3503540">this</span> <span m="3503790">gives</span> <span m="3504340">a</span> <span m="3505300">tournament</span> <span m="3505730">graph.</span> <span m="3509490">So</span> <span m="3509680">essentially,</span> <span m="3510290">so</span> <span m="3510480">far,</span> <span m="3511290">we</span> <span m="3511550">really</span> <span m="3511810">haven't</span> <span m="3512660">done</span> <span m="3512880">anything</span> <span m="3513880">creative</span> <span m="3514750">or</span> <span m="3514920">anything</span> <span m="3515660">that</span> <span m="3515800">we</span> <span m="3516000">had</span> <span m="3516250">to</span> <span m="3516360">make</span> <span m="3516600">a</span> <span m="3516650">big</span> <span m="3516900">leap</span> <span m="3517240">in</span> <span m="3517430">order</span> <span m="3517600">to</span> <span m="3517720">prove</span> <span m="3517990">this</span> <span m="3518220">theorem.</span> </p>
<p><span m="3520420">We</span> <span m="3520520">started</span> <span m="3520900">out</span> <span m="3521180">with,</span> <span m="3521620">if</span> <span m="3521920">you</span> <span m="3522040">want</span> <span m="3522200">to</span> <span m="3522280">prove</span> <span m="3522550">something</span> <span m="3522890">like</span> <span m="3523080">this</span> <span m="3523250">you</span> <span m="3523360">have</span> <span m="3523540">to</span> <span m="3523700">really</span> <span m="3523980">look</span> <span m="3524160">at</span> <span m="3524210">the</span> <span m="3524280">number</span> <span m="3524550">of</span> <span m="3524650">vertices.</span> <span m="3525770">And</span> <span m="3525960">then</span> <span m="3526110">we</span> <span m="3526210">start</span> <span m="3526840">to</span> <span m="3527840">write</span> <span m="3528050">down</span> <span m="3528270">this</span> <span m="3528440">stuff</span> <span m="3528730">over</span> <span m="3528850">here</span> <span m="3529140">that</span> <span m="3529250">makes</span> <span m="3529590">total</span> <span m="3529880">sense.</span> <span m="3530290">And</span> <span m="3530410">then</span> <span m="3531570">we</span> <span m="3531840">are</span> <span m="3532000">going</span> <span m="3532350">to</span> <span m="3533290">figure</span> <span m="3533570">out</span> <span m="3533720">where</span> <span m="3533880">we</span> <span m="3534000">can</span> <span m="3534220">use</span> <span m="3534480">this</span> <span m="3534650">induction</span> <span m="3535140">step.</span> <span m="3535720">And</span> <span m="3535890">so</span> <span m="3536040">we</span> <span m="3536130">just</span> <span m="3536370">take</span> <span m="3536580">out</span> <span m="3536760">one</span> <span m="3536880">node.</span> <span m="3537150">And</span> <span m="3537270">yes,</span> <span m="3537470">it</span> <span m="3537590">is</span> <span m="3537720">a</span> <span m="3537790">tournament</span> <span m="3538180">graph</span> <span m="3541090">on</span> <span m="3541310">n</span> <span m="3541530">nodes.</span> <span m="3544180">So</span> <span m="3544240">this</span> <span m="3544430">is</span> <span m="3544560">very</span> <span m="3544770">systematic.</span> <span m="3545630">That's</span> <span m="3545800">what</span> <span m="3545930">I try</span> <span m="3546110">to get</span> <span m="3546385">at</span> <span m="3546660">here.</span> </p>
<p><span m="3547850">So</span> <span m="3548140">by the</span> <span m="3548370">induction</span> <span m="3548860">step</span> <span m="3549700">because</span> <span m="3550550">by</span> <span m="3550800">the</span> <span m="3550890">induction</span> <span m="3551080">hypothesis,</span> <span m="3554060">we</span> <span m="3554220">know</span> <span m="3554420">now</span> <span m="3555120">that</span> <span m="3555290">we</span> <span m="3555480">actually</span> <span m="3555960">have</span> <span m="3557000">a</span> <span m="3559290">directed</span> <span m="3559660">Hamiltonian</span> <span m="3560120">path.</span> <span m="3561070">So</span> <span m="3561290">let</span> <span m="3562420">V1</span> <span m="3562990">to</span> <span m="3563200">V2</span> <span m="3566010">to</span> <span m="3566340">Vn</span> <span m="3567080">be</span> <span m="3567220">such</span> <span m="3567500">a</span> <span m="3567570">path.</span> <span m="3571170">So</span> <span m="3571340">now that</span> <span m="3571650">we</span> <span m="3571740">can</span> <span m="3571910">use</span> <span m="3572230">this, we</span> <span m="3572700">apply</span> <span m="3573110">it</span> <span m="3574160">and</span> <span m="3574320">we</span> <span m="3574410">get</span> <span m="3574590">a</span> <span m="3574680">path.</span> <span m="3575230">Now</span> <span m="3576100">what</span> <span m="3576290">do</span> <span m="3576380">we</span> <span m="3576520">want</span> <span m="3576700">to</span> <span m="3576800">do?</span> <span m="3577030">We</span> <span m="3577040">want</span> <span m="3577190">to</span> <span m="3577290">show</span> <span m="3577660">that</span> <span m="3577790">we can</span> <span m="3578190">create</span> <span m="3578840">a</span> <span m="3578880">new</span> <span m="3579220">path,</span> <span m="3580510">which</span> <span m="3580670">is</span> <span m="3580780">also</span> <span m="3581070">a</span> <span m="3581270">directed</span> <span m="3582040">Hamiltonian path,</span> <span m="3582990">but</span> <span m="3583130">now</span> <span m="3583390">one</span> <span m="3583610">that</span> <span m="3583780">also</span> <span m="3584140">includes</span> <span m="3584590">V</span> <span m="3584930">in</span> <span m="3585020">the</span> <span m="3585110">bigger</span> <span m="3585360">graph.</span> <span m="3586460">If</span> <span m="3586610">we</span> <span m="3586700">can</span> <span m="3586850">do</span> <span m="3587010">that,</span> <span m="3587300">we</span> <span m="3587390">are</span> <span m="3587450">done.</span> <span m="3588020">So</span> <span m="3588240">now</span> <span m="3588440">we</span> <span m="3588540">have</span> <span m="3588730">to</span> <span m="3588840">start</span> <span m="3589590">really</span> <span m="3589860">looking</span> <span m="3590220">at</span> <span m="3591530">how</span> <span m="3591670">we</span> <span m="3591780">can</span> <span m="3591930">make</span> <span m="3592110">that</span> <span m="3592240">happen.</span> </p>
<p><span m="3599590">In</span> <span m="3599620">order</span> <span m="3599920">to</span> <span m="3600030">do</span> <span m="3600230">this,</span> <span m="3603970">we</span> <span m="3604100">have</span> <span m="3604280">to</span> <span m="3604410">see</span> <span m="3605260">how</span> <span m="3605530">we</span> <span m="3605690">can</span> <span m="3605880">somehow</span> <span m="3606550">plug</span> <span m="3607310">V,</span> <span m="3608090">the</span> <span m="3608230">vertex</span> <span m="3608590">that</span> <span m="3608890">we</span> <span m="3609030">have</span> <span m="3609330">removed,</span> <span m="3610120">in</span> <span m="3610350">this</span> <span m="3610630">path</span> <span m="3610970">over</span> <span m="3611250">here.</span> <span m="3611640">If</span> <span m="3611770">you</span> <span m="3611860">can</span> <span m="3612010">do</span> <span m="3612630">this,</span> <span m="3613530">we</span> <span m="3613740">are</span> <span m="3614250">in</span> <span m="3614640">really</span> <span m="3614890">good</span> <span m="3615050">shape.</span> <span m="3618080">So</span> <span m="3618360">far,</span> <span m="3618650">we</span> <span m="3618770">haven't</span> <span m="3619190">used</span> <span m="3619540">at</span> <span m="3619890">all--</span> <span m="3620110">so</span> <span m="3620430">there's</span> <span m="3620590">also</span> <span m="3620800">something</span> <span m="3621200">that</span> <span m="3621360">you</span> <span m="3621470">can</span> <span m="3621650">look</span> <span m="3621910">at</span> <span m="3622000">if</span> <span m="3622140">you</span> <span m="3622230">start</span> <span m="3622580">solving</span> <span m="3623070">these types of</span> <span m="3623530">problems--</span> <span m="3626710">we</span> <span m="3627110">haven't</span> <span m="3627220">used</span> <span m="3627530">at</span> <span m="3627780">all</span> <span m="3628240">the</span> <span m="3628550">property</span> <span m="3629110">that</span> <span m="3629200">the</span> <span m="3629320">tournament</span> <span m="3629700">graph</span> <span m="3630050">has,</span> <span m="3630430">which</span> <span m="3630680">I</span> <span m="3630900">just</span> <span m="3631190">wiped</span> <span m="3631460">out.</span> <span m="3633900">So</span> <span m="3633980">let's</span> <span m="3634140">figure</span> <span m="3634510">out</span> <span m="3634600">what</span> <span m="3634875">we</span> <span m="3635150">can</span> <span m="3635540">do</span> <span m="3635700">here.</span> <span m="3638870">We</span> <span m="3638950">should</span> <span m="3639080">be</span> <span m="3639210">able</span> <span m="3639330">to</span> <span m="3639440">use</span> <span m="3639620">something.</span> </p>
<p><span m="3641070">So</span> <span m="3641270">of</span> <span m="3641420">course,</span> <span m="3642040">we</span> <span m="3642110">have</span> <span m="3642230">a</span> <span m="3642280">few</span> <span m="3642460">simple</span> <span m="3642860">cases.</span> <span m="3643740">For</span> <span m="3643920">example,</span> <span m="3644820">if</span> <span m="3645220">V</span> <span m="3645580">has</span> <span m="3646020">a</span> <span m="3646370">directed</span> <span m="3646960">edge</span> <span m="3647340">into</span> <span m="3647710">V1,</span> <span m="3651170">and</span> <span m="3651580">I</span> <span m="3651730">have</span> <span m="3651900">a</span> <span m="3651960">Hamiltonian</span> <span m="3652930">path</span> <span m="3653170">that</span> <span m="3653340">goes</span> <span m="3653570">from</span> <span m="3653700">V</span> <span m="3654030">to</span> <span m="3654150">V1</span> <span m="3655370">to</span> <span m="3655500">V2</span> <span m="3655970">all</span> <span m="3656130">the</span> <span m="3656220">way</span> <span m="3656350">to</span> <span m="3656580">Vn,</span> <span m="3657210">and</span> <span m="3657330">then</span> <span m="3657460">I</span> <span m="3657530">cover</span> <span m="3658000">all</span> <span m="3658220">the</span> <span m="3658320">vertices</span> <span m="3659080">exactly</span> <span m="3659500">once.</span> <span m="3660610">And</span> <span m="3660840">I</span> <span m="3660870">will</span> <span m="3661020">have</span> <span m="3661230">a</span> <span m="3661360">direct</span> <span m="3661740">Hamiltonian</span> <span m="3662360">path.</span> <span m="3663540">So</span> <span m="3663710">this</span> <span m="3663950">is</span> <span m="3664110">great.</span> <span m="3664480">So</span> <span m="3664690">this</span> <span m="3664940">is</span> <span m="3665180">definitely</span> <span m="3665840">easy.</span> <span m="3667420">So</span> <span m="3669140">this</span> <span m="3669330">is</span> <span m="3669670">case</span> <span m="3669960">one.</span> </p>
<p><span m="3670615">In</span> <span m="3670890">case</span> <span m="3671270">two,</span> <span m="3673500">suppose</span> <span m="3674210">that</span> <span m="3674310">V1</span> <span m="3675110">has</span> <span m="3675520">a</span> <span m="3675620">directed</span> <span m="3675980">edge</span> <span m="3676590">to</span> <span m="3676980">V.</span> <span m="3679300">And</span> <span m="3680730">So</span> <span m="3680950">now</span> <span m="3681230">we</span> <span m="3681350">have</span> <span m="3681630">a</span> <span m="3681680">little</span> <span m="3681910">problem</span> <span m="3683510">because</span> <span m="3684010">somehow</span> <span m="3684610">there</span> <span m="3684790">is</span> <span m="3684920">an</span> <span m="3685070">edge</span> <span m="3687200">like</span> <span m="3687400">this</span> <span m="3687680">to</span> <span m="3687770">V,</span> <span m="3688100">but</span> <span m="3689250">they</span> <span m="3689350">cannot</span> <span m="3689730">go</span> <span m="3689920">like</span> <span m="3690110">this.</span> <span m="3690470">This</span> <span m="3690600">is not</span> <span m="3690770">a</span> <span m="3690850">Hamiltonian</span> <span m="3691410">path.</span> <span m="3691870">I</span> <span m="3691940">need</span> <span m="3692120">to</span> <span m="3692240">have</span> <span m="3692370">a</span> <span m="3692430">directed</span> <span m="3692980">walk</span> <span m="3695000">that</span> <span m="3695180">covers</span> <span m="3695530">all</span> <span m="3695750">the</span> <span m="3695830">vertices</span> <span m="3696290">exactly</span> <span m="3696380">once.</span> <span m="3696980">So</span> <span m="3697130">now</span> <span m="3697320">we</span> <span m="3697400">have</span> <span m="3697570">a</span> <span m="3697630">little</span> <span m="3697830">problem.</span> </p>
<p><span m="3698690">So</span> <span m="3698840">now</span> <span m="3699050">we</span> <span m="3699140">have</span> <span m="3699270">to</span> <span m="3699370">start</span> <span m="3699690">really</span> <span m="3699970">thinking</span> <span m="3700400">about</span> <span m="3700660">how</span> <span m="3700860">we</span> <span m="3700980">can</span> <span m="3701500">solve</span> <span m="3701900">this.</span> <span m="3703550">So</span> <span m="3703770">are</span> <span m="3703860">there</span> <span m="3703950">any</span> <span m="3704150">suggestions</span> <span m="3704880">to</span> <span m="3704990">make</span> <span m="3705190">this</span> <span m="3705380">happen?</span> <span m="3707580">So</span> <span m="3707660">let's</span> <span m="3707900">think</span> <span m="3708120">a</span> <span m="3708150">little</span> <span m="3708370">bit</span> <span m="3708530">about</span> <span m="3708880">this.</span> <span m="3712760">So</span> <span m="3712910">somehow</span> <span m="3713910">if</span> <span m="3714110">I</span> <span m="3714140">start</span> <span m="3714430">thinking</span> <span m="3714730">about</span> <span m="3714980">this,</span> <span m="3715160">I</span> <span m="3715220">would</span> <span m="3715390">like</span> <span m="3715580">to</span> <span m="3716550">plug</span> <span m="3716990">V</span> <span m="3717790">somewhere</span> <span m="3718340">in</span> <span m="3718530">this</span> <span m="3718650">sequence.</span> <span m="3719450">That</span> <span m="3719610">will</span> <span m="3719720">be</span> <span m="3720060">like</span> <span m="3720370">a</span> <span m="3720570">pretty</span> <span m="3720860">obvious</span> <span m="3721195">way to--</span> <span m="3721760">go</span> <span m="3722070">ahead.</span> </p>
<p><span m="3723495">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3736270">PROFESSOR: Yeah.</span> <span m="3737015">That's</span> <span m="3737490">true.</span> <span m="3738440">So for</span> <span m="3738915">example,</span> <span m="3739870">I</span> <span m="3740010">may</span> <span m="3740340">have</span> <span m="3740750">that</span> <span m="3742420">for</span> <span m="3742710">example,</span> <span m="3743160">V2</span> <span m="3743890">beats</span> <span m="3744480">V</span> <span m="3744950">as</span> <span m="3745210">well.</span> <span m="3747110">But</span> <span m="3747260">suppose</span> <span m="3747960">that</span> <span m="3748140">I</span> <span m="3748200">have</span> <span m="3748440">V3</span> <span m="3749010">over</span> <span m="3749350">here,</span> <span m="3749860">and</span> <span m="3750030">this</span> <span m="3750230">one</span> <span m="3750430">beats</span> <span m="3751060">V3.</span> <span m="3752820">Then,</span> <span m="3753560">as</span> <span m="3753730">you say, I</span> <span m="3754160">could</span> <span m="3754330">sort</span> <span m="3754530">of</span> <span m="3754620">plug</span> <span m="3754960">V</span> <span m="3755240">in</span> <span m="3755420">here</span> <span m="3755790">in</span> <span m="3755940">this</span> <span m="3756140">sequence</span> <span m="3756670">and</span> <span m="3756760">I</span> <span m="3756780">may</span> <span m="3756990">have</span> <span m="3757190">a</span> <span m="3757240">longer</span> <span m="3757500">sequence.</span> <span m="3758860">Or</span> <span m="3759130">if</span> <span m="3759220">this</span> <span m="3759400">is</span> <span m="3759490">not</span> <span m="3759650">the</span> <span m="3759760">case,</span> <span m="3760940">then</span> <span m="3762530">maybe</span> <span m="3762870">the</span> <span m="3762970">next</span> <span m="3763350">one,</span> <span m="3764800">V4,</span> <span m="3766250">may</span> <span m="3766520">have</span> <span m="3766770">the</span> <span m="3766890">property</span> <span m="3767200">and</span> <span m="3767510">I can</span> <span m="3767900">plug V</span> <span m="3768290">in</span> <span m="3768500">here.</span> <span m="3769450">So</span> <span m="3769600">essentially</span> <span m="3770000">what</span> <span m="3770150">I</span> <span m="3770170">want</span> <span m="3770340">to</span> <span m="3770440">show</span> <span m="3770870">is</span> <span m="3773335">that</span> <span m="3773680">I</span> <span m="3773790">can</span> <span m="3774030">plug</span> <span m="3774360">V</span> <span m="3774620">somewhere</span> <span m="3775360">in</span> <span m="3776310">the</span> <span m="3776430">middle</span> <span m="3777000">of</span> <span m="3777610">two</span> <span m="3778080">of</span> <span m="3778210">these</span> <span m="3779050">Vi's.</span> </p>
<p><span m="3780200">And</span> <span m="3781280">the</span> <span m="3781410">property</span> <span m="3781820">that</span> <span m="3782000">you</span> <span m="3782170">were using</span> <span m="3782490">is</span> <span m="3782750">actually</span> <span m="3783190">that</span> <span m="3783360">you</span> <span m="3783460">said,</span> <span m="3783860">well,</span> <span m="3784300">I</span> <span m="3784420">know</span> <span m="3785200">that</span> <span m="3786520">V2</span> <span m="3786770">either</span> <span m="3787040">beats</span> <span m="3787450">V,</span> <span m="3787860">or</span> <span m="3788400">V beats</span> <span m="3788843">V2.</span> <span m="3789730">So</span> <span m="3789870">that's</span> <span m="3790300">the</span> <span m="3790490">property</span> <span m="3790860">of</span> <span m="3790950">the</span> <span m="3791050">tournament</span> <span m="3791450">graph</span> <span m="3792320">that</span> <span m="3792490">we're going</span> <span m="3792690">to</span> <span m="3792830">use</span> <span m="3793140">here</span> <span m="3793320">in</span> <span m="3793420">order</span> <span m="3793580">to</span> <span m="3793710">prove</span> <span m="3794040">this.</span> <span m="3795600">So</span> <span m="3796060">yes,</span> <span m="3796270">that's a</span> <span m="3796500">great</span> <span m="3796870">observation.</span> </p>
<p><span m="3799430">So</span> <span m="3799830">the</span> <span m="3799930">way</span> <span m="3800070">you</span> <span m="3800230">formulated</span> <span m="3800890">it,</span> <span m="3801110">we</span> <span m="3801280">can</span> <span m="3801540">maybe</span> <span m="3802300">use</span> <span m="3802660">induction,</span> <span m="3803220">for</span> <span m="3803410">example,</span> <span m="3803930">to</span> <span m="3804040">prove</span> <span m="3804400">this.</span> <span m="3804770">We</span> <span m="3804870">can</span> <span m="3808610">sort</span> <span m="3808810">of</span> <span m="3808930">go</span> <span m="3809370">recursively</span> <span m="3810060">through</span> <span m="3810410">this</span> <span m="3810900">until</span> <span m="3811920">you</span> <span m="3812080">find</span> <span m="3812400">the</span> <span m="3812500">right</span> <span m="3812740">spot.</span> <span m="3813850">Maybe</span> <span m="3813990">they</span> <span m="3814270">can</span> <span m="3814700">immediately</span> <span m="3815790">precisely</span> <span m="3816440">indicate</span> <span m="3816950">such</span> <span m="3817210">a</span> <span m="3817260">spot.</span> <span m="3818390">So</span> <span m="3818590">how</span> <span m="3818750">do</span> <span m="3818890">we</span> <span m="3818990">usually</span> <span m="3819360">do</span> <span m="3819520">that?</span> <span m="3819750">If</span> <span m="3819890">we</span> <span m="3820030">are</span> <span m="3820730">thinking</span> <span m="3821220">about</span> <span m="3822916">other</span> <span m="3823330">theorems</span> <span m="3823690">that</span> <span m="3823870">we</span> <span m="3823980">have</span> <span m="3824090">tried</span> <span m="3824380">to</span> <span m="3824500">prove.</span> </p>
<p><span m="3831850">AUDIENCE: What's</span> <span m="3832040">the</span> <span m="3832120">smallest</span> <span m="3832580">value</span> <span m="3833020">of</span> <span m="3833360">i such</span> <span m="3833720">that</span> <span m="3834280">V</span> <span m="3834655">beats</span> <span m="3835030">Vi?</span> </p>
<p><span m="3836652">PROFESSOR: OK.</span> <span m="3837600">So what's</span> <span m="3837990">the smallest</span> <span m="3839410">value</span> <span m="3839900">of</span> <span m="3840390">i</span> <span m="3840880">where</span> <span m="3841370">V beats</span> <span m="3841860">Vi.</span> <span m="3854930">So</span> <span m="3855070">usually</span> <span m="3855500">we</span> <span m="3855620">have</span> <span m="3855800">words</span> <span m="3856180">in</span> <span m="3856300">our</span> <span m="3856550">mind</span> <span m="3856910">like</span> <span m="3857530">largest</span> <span m="3858300">or</span> <span m="3858470">smallest,</span> <span m="3861780">et</span> <span m="3862090">cetera,</span> <span m="3862610">and</span> <span m="3863840">sort</span> <span m="3864050">of</span> <span m="3864290">an</span> <span m="3864430">extreme</span> <span m="3864830">precision.</span> <span m="3865440">And</span> <span m="3865540">then</span> <span m="3865630">we</span> <span m="3865740">like</span> <span m="3865930">to</span> <span m="3866050">find</span> <span m="3866390">out</span> <span m="3866540">that</span> <span m="3866720">we</span> <span m="3866850">can</span> <span m="3867000">make</span> <span m="3867210">it</span> <span m="3867320">happen.</span> <span m="3867690">And</span> <span m="3867770">then</span> <span m="3867890">we</span> <span m="3867990">say,</span> <span m="3868900">well,</span> <span m="3870570">if</span> <span m="3871460">something</span> <span m="3871800">goes</span> <span m="3872080">wrong,</span> <span m="3872880">we</span> <span m="3873050">violate</span> <span m="3873670">that</span> <span m="3873780">smallest</span> <span m="3874280">condition.</span> <span m="3876090">So</span> <span m="3876240">let's</span> <span m="3876630">do</span> <span m="3876850">that</span> <span m="3877020">here.</span> <span m="3877300">So</span> <span m="3877440">let's</span> <span m="3877650">indicate</span> <span m="3878120">a</span> <span m="3878180">specific</span> <span m="3878740">spot.</span> </p>
<p><span m="3881380">Let's</span> <span m="3881630">considered</span> <span m="3882110">the</span> <span m="3882240">smallest</span> <span m="3884476">i</span> <span m="3886410">such</span> <span m="3886430">that</span> <span m="3892230">V</span> <span m="3893820">beats</span> <span m="3894500">Vi.</span> <span m="3898860">Well,</span> <span m="3899020">let's</span> <span m="3899100">have a</span> <span m="3899410">look</span> <span m="3899900">how</span> <span m="3900080">this</span> <span m="3900280">would</span> <span m="3900430">work.</span> <span m="3904030">So</span> <span m="3904600">this</span> <span m="3904820">is</span> <span m="3905000">a</span> <span m="3905380">little</span> <span m="3905600">bit</span> <span m="3905790">of</span> <span m="3905930">a</span> <span m="3906190">different</span> <span m="3907420">proof</span> <span m="3907760">than</span> <span m="3908270">what</span> <span m="3908460">I</span> <span m="3908620">had</span> <span m="3908830">but</span> <span m="3909870">this</span> <span m="3909980">should</span> <span m="3910320">work</span> <span m="3910550">fine</span> <span m="3910830">as</span> <span m="3910960">well.</span> <span m="3911250">So</span> <span m="3911270">let's</span> <span m="3911500">just</span> <span m="3911650">see</span> <span m="3911790">how</span> <span m="3911930">it</span> <span m="3912030">works.</span> </p>
<p><span m="3916210">So</span> <span m="3916340">we</span> <span m="3916450">have</span> <span m="3916830">V1.</span> <span m="3922640">First</span> <span m="3922940">of</span> <span m="3922990">all,</span> <span m="3923250">we</span> <span m="3923360">notice</span> <span m="3924110">that,</span> <span m="3924490">of</span> <span m="3924650">course,</span> <span m="3924980">if</span> <span m="3925500">i</span> <span m="3925730">equals</span> <span m="3926140">1,</span> <span m="3926650">which</span> <span m="3928220">that</span> <span m="3928360">cannot</span> <span m="3928610">be</span> <span m="3928710">the</span> <span m="3928840">case,</span> <span m="3929200">so</span> <span m="3929330">we</span> <span m="3929430">know that</span> <span m="3929740">i is</span> <span m="3930110">larger</span> <span m="3930970">than</span> <span m="3931180">1.</span> <span m="3931850">So</span> <span m="3932060">there's</span> <span m="3932300">really</span> <span m="3933600">somewhere</span> <span m="3934630">a</span> <span m="3934940">Vi</span> <span m="3937030">minus</span> <span m="3937590">1--</span> <span m="3938310">we have</span> <span m="3938550">to</span> <span m="3938660">check</span> <span m="3938940">that,</span> <span m="3939140">right?</span> <span m="3939560">If that</span> <span m="3939970">exists,</span> <span m="3941050">this</span> <span m="3941240">index</span> <span m="3941560">is</span> <span m="3941650">not</span> <span m="3941780">equal</span> <span m="3941950">to</span> <span m="3942030">zero--</span> <span m="3942700">that</span> <span m="3942850">goes</span> <span m="3943110">to</span> <span m="3943310">Vi</span> <span m="3944540">and</span> <span m="3944710">then</span> <span m="3944890">goes</span> <span m="3945260">all</span> <span m="3945490">the</span> <span m="3945580">way</span> <span m="3945840">up</span> <span m="3946050">to</span> <span m="3949080">Vn.</span> </p>
<p><span m="3954580">So</span> <span m="3954780">now</span> <span m="3955030">we</span> <span m="3955170">say</span> <span m="3957220">we</span> <span m="3957490">can</span> <span m="3957730">use</span> <span m="3958260">this,</span> <span m="3959690">that</span> <span m="3960005">V</span> <span m="3960320">actually</span> <span m="3960690">beats</span> <span m="3960920">Vi.</span> <span m="3963680">Now</span> <span m="3963920">what's</span> <span m="3964120">about--</span> <span m="3966030">maybe</span> <span m="3966280">someone</span> <span m="3966600">else</span> <span m="3966800">can</span> <span m="3966910">help</span> <span m="3967200">me</span> <span m="3967350">here--</span> <span m="3969870">I</span> <span m="3970010">would</span> <span m="3970290">like</span> <span m="3970950">to</span> <span m="3971170">have</span> <span m="3971480">that</span> <span m="3971640">Vi</span> <span m="3972120">minus</span> <span m="3972600">1</span> <span m="3973285">beats</span> <span m="3973550">V.</span> <span m="3975190">That</span> <span m="3975340">would</span> <span m="3975450">be</span> <span m="3975580">fantastic</span> <span m="3976770">because</span> <span m="3977150">then</span> <span m="3978020">I</span> <span m="3978210">have</span> <span m="3978630">a</span> <span m="3978750">path</span> <span m="3979030">that</span> <span m="3979310">goes</span> <span m="3979590">from</span> <span m="3979930">V1</span> <span m="3980120">all the</span> <span m="3980370">way</span> <span m="3980580">up</span> <span m="3980840">here,</span> <span m="3981300">goes</span> <span m="3981600">here,</span> <span m="3982220">goes</span> <span m="3982480">there,</span> <span m="3982880">and</span> <span m="3983060">all</span> <span m="3983220">the</span> <span m="3983310">way</span> <span m="3983500">up</span> <span m="3983610">to</span> <span m="3983740">here.</span> <span m="3984470">So</span> <span m="3984690">we</span> <span m="3984830">have</span> <span m="3985030">a</span> <span m="3985110">directed</span> <span m="3985440">Hamiltonian</span> <span m="3986230">path</span> <span m="3986535">that</span> <span m="3986840">covers</span> <span m="3987950">all</span> <span m="3988100">the</span> <span m="3988210">vertices</span> <span m="3989060">exactly</span> <span m="3989610">once,</span> <span m="3989950">and</span> <span m="3990020">that's</span> <span m="3990200">what</span> <span m="3990330">you</span> <span m="3990470">want</span> <span m="3990620">to</span> <span m="3990700">prove.</span> <span m="3992210">But</span> <span m="3992380">why</span> <span m="3992570">would</span> <span m="3992740">there</span> <span m="3992840">be</span> <span m="3993060">an</span> <span m="3993170">edge</span> <span m="3993480">that</span> <span m="3993670">goes</span> <span m="3993930">this</span> <span m="3994120">way?</span> <span m="3997010">So</span> <span m="3997160">how</span> <span m="3997310">do</span> <span m="3997400">we</span> <span m="3997540">reason</span> <span m="3997750">about this?</span> <span m="4004390">Someone</span> <span m="4004690">else?</span> </p>
<p><span m="4006300">AUDIENCE: Because</span> <span m="4006840">Vi</span> <span m="4007050">is the</span> <span m="4007380">smallest</span> <span m="4008860">number</span> <span m="4009260">that</span> <span m="4009860">V</span> <span m="4011580">beats,</span> <span m="4012270">then</span> <span m="4012660">anything</span> <span m="4013030">smaller</span> <span m="4013460">than</span> <span m="4013927">i</span> <span m="4014394">must</span> <span m="4014861">have beat</span> <span m="4015328">V.</span> </p>
<p><span m="4016729">PROFESSOR: Exactly.</span> <span m="4018780">So</span> <span m="4019860">if</span> <span m="4020570">V</span> <span m="4021190">would</span> <span m="4021300">beat</span> <span m="4021640">Vi</span> <span m="4022350">minus</span> <span m="4022800">1,</span> <span m="4023910">that</span> <span m="4024110">would</span> <span m="4024260">contradict</span> <span m="4024910">the</span> <span m="4025010">smallest</span> <span m="4025590">property</span> <span m="4026040">over</span> <span m="4026310">here.</span> <span m="4027130">So</span> <span m="4027270">that's</span> <span m="4027410">a</span> <span m="4027470">contradiction.</span> <span m="4029010">Now we</span> <span m="4029300">use</span> <span m="4029580">a</span> <span m="4029670">property</span> <span m="4030250">of</span> <span m="4030350">the</span> <span m="4030510">tournament</span> <span m="4030950">graph.</span> <span m="4032070">So</span> <span m="4032270">now</span> <span m="4032450">I</span> <span m="4032550">know</span> <span m="4033390">that</span> <span m="4033560">Vi</span> <span m="4034040">minus</span> <span m="4034530">1</span> <span m="4034990">must</span> <span m="4035270">beat</span> <span m="4035500">V.</span> <span m="4036150">So</span> <span m="4036370">I</span> <span m="4036470">really</span> <span m="4036780">have</span> <span m="4037000">an</span> <span m="4037120">edge</span> <span m="4037420">over</span> <span m="4037700">here.</span> <span m="4039920">So</span> <span m="4040130">that</span> <span m="4040350">works.</span> <span m="4041450">So</span> <span m="4041670">that's</span> <span m="4042040">the</span> <span m="4042480">end</span> <span m="4042690">of</span> <span m="4042820">the</span> <span m="4042910">proof.</span> </p>
<p><span m="4044040">Now</span> <span m="4044320">another</span> <span m="4045120">version</span> <span m="4045760">could</span> <span m="4045930">have</span> <span m="4046150">been</span> <span m="4047480">in</span> <span m="4047710">which</span> <span m="4048760">we</span> <span m="4048880">would</span> <span m="4049020">do</span> <span m="4049130">something</span> <span m="4049500">similar</span> <span m="4049940">like</span> <span m="4050170">this.</span> <span m="4051280">But</span> <span m="4051450">we</span> <span m="4051540">could</span> <span m="4051740">also</span> <span m="4052060">have</span> <span m="4052290">used,</span> <span m="4052470">say,</span> <span m="4052800">the</span> <span m="4052970">largest</span> <span m="4053250">i--</span> <span m="4055320">just</span> <span m="4056590">something</span> <span m="4057060">that</span> <span m="4057160">you</span> <span m="4057300">may</span> <span m="4057460">want</span> <span m="4057670">to</span> <span m="4058060">look</span> <span m="4058310">at.</span> <span m="4058480">You</span> <span m="4058610">can</span> <span m="4058750">also</span> <span m="4058950">use</span> <span m="4059540">the</span> <span m="4059940">largest</span> <span m="4060470">i</span> <span m="4061180">such</span> <span m="4061580">that</span> <span m="4064500">Vi</span> <span m="4065960">beats</span> <span m="4066440">V.</span> <span m="4067740">We</span> <span m="4067950">have a</span> <span m="4068000">completely</span> <span m="4068480">symmetrical</span> <span m="4069130">argument</span> <span m="4069660">here,</span> <span m="4070320">but</span> <span m="4070490">you</span> <span m="4070600">could</span> <span m="4070880">use</span> <span m="4071180">this</span> <span m="4071340">solution</span> <span m="4071770">as</span> <span m="4071900">well.</span> <span m="4072630">So</span> <span m="4072960">I'm</span> <span m="4073040">just</span> <span m="4073270">trying</span> <span m="4073490">to</span> <span m="4073620">sketch</span> <span m="4074040">here</span> <span m="4074570">the</span> <span m="4074680">way</span> <span m="4074860">of</span> <span m="4075000">thinking</span> <span m="4075330">that</span> <span m="4075480">you</span> <span m="4075600">may</span> <span m="4075790">want</span> <span m="4075980">to</span> <span m="4076090">consider</span> <span m="4076920">in</span> <span m="4077030">these</span> <span m="4077270">types</span> <span m="4077530">of</span> <span m="4077610">problems.</span> </p>
<p><span m="4078550">So</span> <span m="4078720">why</span> <span m="4078860">would</span> <span m="4078980">this</span> <span m="4079180">work,</span> <span m="4079390">by</span> <span m="4079540">the</span> <span m="4079660">way?</span> <span m="4080220">Well,</span> <span m="4080460">we have the</span> <span m="4080530">same</span> <span m="4080850">kind</span> <span m="4081070">of</span> <span m="4081190">argument</span> <span m="4081650">like</span> <span m="4081850">this.</span> <span m="4082780">We</span> <span m="4083530">plug</span> <span m="4083820">V</span> <span m="4084470">right</span> <span m="4085140">after</span> <span m="4085620">Vi.</span> <span m="4086610">We</span> <span m="4086740">know</span> <span m="4086900">there's</span> <span m="4087160">an</span> <span m="4087300">edge</span> <span m="4087660">from</span> <span m="4087990">V to</span> <span m="4088320">Vi</span> <span m="4088480">plus</span> <span m="4088800">1.</span> <span m="4089560">Why?</span> <span m="4090340">If</span> <span m="4090530">it's</span> <span m="4090730">not</span> <span m="4090890">the</span> <span m="4091000">case,</span> <span m="4091920">there</span> <span m="4092070">will</span> <span m="4092210">be</span> <span m="4092340">a</span> <span m="4092420">large</span> <span m="4092800">index,</span> <span m="4093280">i,</span> <span m="4094400">that</span> <span m="4094610">contradicts</span> <span m="4095170">our</span> <span m="4095540">assumption</span> <span m="4096189">that</span> <span m="4096399">we</span> <span m="4096490">have</span> <span m="4096689">the</span> <span m="4096790">largest</span> <span m="4097200">i</span> <span m="4097390">already.</span> <span m="4098470">It's</span> <span m="4098630">a</span> <span m="4098729">tournament</span> <span m="4099120">graph,</span> <span m="4099830">so</span> <span m="4100080">we</span> <span m="4100200">know</span> <span m="4100560">that</span> <span m="4100710">there's</span> <span m="4100960">an</span> <span m="4101080">edge</span> <span m="4101970">from</span> <span m="4103149">V</span> <span m="4103450">to Vi</span> <span m="4103970">plus</span> <span m="4104210">1.</span> <span m="4104890">And</span> <span m="4105040">we</span> <span m="4105109">get</span> <span m="4105220">a</span> <span m="4105310">directed</span> <span m="4105490">Hamiltonian</span> <span m="4106800">path as</span> <span m="4107240">well.</span> <span m="4107920">So</span> <span m="4107950">you</span> <span m="4108000">may</span> <span m="4108109">want</span> <span m="4108260">to</span> <span m="4108380">look</span> <span m="4108550">at</span> <span m="4108609">that</span> <span m="4108790">as</span> <span m="4108930">well.</span> </p>
<p><span m="4114729">So</span> <span m="4114850">this</span> <span m="4114970">is</span> <span m="4115220">about</span> <span m="4115670">tournament</span> <span m="4116029">graphs.</span> <span m="4116370">So</span> <span m="4116680">let's</span> <span m="4116930">talk</span> <span m="4117170">about</span> <span m="4118439">an</span> <span m="4118660">interesting</span> <span m="4119740">tournament</span> <span m="4120250">graph</span> <span m="4121439">with</span> <span m="4121640">a</span> <span m="4121729">funny</span> <span m="4122050">game.</span> <span m="4126040">And</span> <span m="4126950">this</span> <span m="4127240">is</span> <span m="4127450">actually</span> <span m="4129439">a</span> <span m="4129550">chicken</span> <span m="4130090">tournament,</span> <span m="4130569">like</span> <span m="4131979">the</span> <span m="4132100">chickens</span> <span m="4132580">here</span> <span m="4133270">represent</span> <span m="4133800">the</span> <span m="4133870">vertices</span> <span m="4134870">and</span> <span m="4135020">they are</span> <span m="4135330">pecking</span> <span m="4135630">one</span> <span m="4135810">another,</span> <span m="4137160">but</span> <span m="4138590">in</span> <span m="4138700">a</span> <span m="4138750">certain</span> <span m="4139229">rule</span> <span m="4139630">that</span> <span m="4139740">defines</span> <span m="4140319">a</span> <span m="4140399">chicken</span> <span m="4140740">to</span> <span m="4140819">be the</span> <span m="4140960">king</span> <span m="4141290">chicken.</span> <span m="4141710">So</span> <span m="4141920">let's</span> <span m="4142109">see</span> <span m="4142229">how</span> <span m="4142380">that</span> <span m="4142550">works.</span> <span m="4147399">So</span> <span m="4147540">that's</span> <span m="4147710">a</span> <span m="4147750">great</span> <span m="4148000">application</span> <span m="4148640">of</span> <span m="4150470">graph</span> <span m="4150779">theory.</span> </p>
<p><span m="4155189">So</span> <span m="4155399">what</span> <span m="4155510">do</span> <span m="4155590">we</span> <span m="4157029">have?</span> <span m="4157210">We have that</span> <span m="4157439">either</span> <span m="4158630">a</span> <span m="4158760">chicken,</span> <span m="4160770">U,</span> <span m="4162370">pecks</span> <span m="4165540">a</span> <span m="4165649">chicken,</span> <span m="4168830">V.</span> <span m="4170130">And</span> <span m="4171350">we</span> <span m="4171500">said</span> <span m="4171830">that</span> <span m="4171960">U</span> <span m="4172310">has a direct</span> <span m="4172700">edge</span> <span m="4172859">to</span> <span m="4173140">V,</span> <span m="4173540">so we're</span> <span m="4173729">actually</span> <span m="4174040">defining</span> <span m="4174470">a</span> <span m="4174540">tournament</span> <span m="4174930">graph</span> <span m="4175250">here.</span> <span m="4176250">Or</span> <span m="4177210">we</span> <span m="4177340">have</span> <span m="4177550">a</span> <span m="4177779">chicken,</span> <span m="4178380">V,</span> <span m="4178859">that</span> <span m="4179020">pecks</span> <span m="4179700">a</span> <span m="4180010">chicken,</span> <span m="4181370">U,</span> <span m="4182500">and</span> <span m="4182670">we</span> <span m="4182819">get</span> <span m="4182979">V</span> <span m="4183630">has a</span> <span m="4184020">direct</span> <span m="4184210">edge</span> <span m="4184569">to</span> <span m="4184710">U.</span> <span m="4184950">So</span> <span m="4185090">we</span> <span m="4185170">have</span> <span m="4185279">a</span> <span m="4185380">tournament</span> <span m="4185770">graph.</span> </p>
<p><span m="4187000">But</span> <span m="4187160">now</span> <span m="4187350">we</span> <span m="4187470">define</span> <span m="4187870">something</span> <span m="4188210">new.</span> <span m="4190680">We</span> <span m="4190920">say</span> <span m="4191220">that</span> <span m="4191740">U</span> <span m="4192220">virtually</span> <span m="4198160">pecks</span> <span m="4198770">V</span> <span m="4201260">if</span> <span m="4201520">one</span> <span m="4201730">of</span> <span m="4201860">the</span> <span m="4201960">two</span> <span m="4202150">conditions</span> <span m="4202660">holds--</span> <span m="4203180">one</span> <span m="4203360">of</span> <span m="4203480">these</span> <span m="4203710">two</span> <span m="4203880">conditions--</span> <span m="4205070">either</span> <span m="4206440">U,</span> <span m="4206710">of</span> <span m="4206850">course,</span> <span m="4207150">pecks</span> <span m="4207240">V.</span> <span m="4208200">That's</span> <span m="4208380">great.</span> <span m="4209950">He's</span> <span m="4210370">in</span> <span m="4210450">good</span> <span m="4210620">shape.</span> <span m="4214520">Or</span> <span m="4215540">there</span> <span m="4215690">exists</span> <span m="4216100">another</span> <span m="4216420">chicken,</span> <span m="4217180">W,</span> <span m="4218290">such</span> <span m="4218660">that</span> <span m="4219430">U</span> <span m="4219900">actually</span> <span m="4220870">pecks</span> <span m="4221297">W</span> <span m="4223210">and</span> <span m="4224740">W,</span> <span m="4225200">in</span> <span m="4225400">turn,</span> <span m="4227760">pecks</span> <span m="4228120">V.</span> <span m="4230490">So</span> <span m="4230650">this</span> <span m="4230880">is</span> <span m="4231060">very</span> <span m="4231260">special</span> <span m="4233960">kind</span> <span m="4234260">of</span> <span m="4234410">first</span> <span m="4234880">relationship.</span> <span m="4235740">So</span> <span m="4236540">we are</span> <span m="4236680">wondering</span> <span m="4237140">now</span> <span m="4244000">can</span> <span m="4244140">we</span> <span m="4244260">now</span> <span m="4244480">define</span> <span m="4244940">something--</span> <span m="4245350">is there</span> <span m="4245630">a</span> <span m="4245750">question?</span> </p>
<p><span m="4246250">AUDIENCE:</span> <span m="4246750">[INAUDIBLE]</span> <span m="4248250">in</span> <span m="4248750">between?</span> <span m="4249250">To</span> <span m="4249600">be</span> <span m="4249890">virtually</span> <span m="4250140">pecked,</span> <span m="4251510">is</span> <span m="4251690">one</span> <span m="4252400">chicken</span> <span m="4252540">in between</span> <span m="4252760">U</span> <span m="4253175">and</span> <span m="4253590">the</span> <span m="4253955">other one?</span> </p>
<p><span m="4256380">PROFESSOR: Well,</span> <span m="4256670">there can be</span> <span m="4256910">multiple</span> <span m="4257500">chickens</span> <span m="4258240">in</span> <span m="4258727">between</span> <span m="4259214">here.</span> <span m="4261649">I</span> <span m="4262136">have several</span> <span m="4262630">friends who</span> <span m="4263120">help</span> <span m="4263370">me</span> <span m="4263550">out</span> <span m="4263770">pecking</span> <span m="4264249">someone else.</span> <span m="4270410">So</span> <span m="4270640">when</span> <span m="4270840">we</span> <span m="4270950">were</span> <span m="4271100">looking</span> <span m="4271590">at</span> <span m="4272930">these</span> <span m="4273620">tournament</span> <span m="4274200">graphs,</span> <span m="4275080">we were</span> <span m="4275230">wondering,</span> <span m="4276520">can</span> <span m="4277040">we</span> <span m="4277340">really</span> <span m="4277600">indicate</span> <span m="4278180">a</span> <span m="4278740">winning</span> <span m="4280480">player?</span> <span m="4281520">Well</span> <span m="4281740">in</span> <span m="4281870">this</span> <span m="4282150">case,</span> <span m="4282460">if</span> <span m="4282580">you</span> <span m="4282670">start</span> <span m="4282990">to</span> <span m="4283100">talk</span> <span m="4283390">about</span> <span m="4283650">virtual</span> <span m="4284120">pecking,</span> <span m="4284440">we</span> <span m="4284620">look</span> <span m="4284810">at</span> <span m="4284870">the</span> <span m="4284970">pecking</span> <span m="4285290">order.</span> <span m="4286580">Then</span> <span m="4288710">we</span> <span m="4288840">can</span> <span m="4289020">actually</span> <span m="4289730">define</span> <span m="4290180">something</span> <span m="4291040">like</span> <span m="4292260">a</span> <span m="4292350">chicken</span> <span m="4292740">king.</span> <span m="4293350">So</span> <span m="4294360">let</span> <span m="4294560">me</span> <span m="4295620">write</span> <span m="4295900">that</span> <span m="4296250">down.</span> <span m="4302986">Let me</span> <span m="4303420">first</span> <span m="4305000">explain</span> <span m="4305400">what</span> <span m="4305540">I</span> <span m="4305590">mean</span> <span m="4305840">here.</span> <span m="4307040">And I</span> <span m="4307230">give</span> <span m="4307350">an</span> <span m="4307440">example</span> <span m="4308000">of</span> <span m="4308090">a</span> <span m="4308120">graph.</span> </p>
<p><span m="4319260">So</span> <span m="4319500">a</span> <span m="4319570">chicken</span> <span m="4321470">that</span> <span m="4321660">is</span> <span m="4321890">able</span> <span m="4322340">to</span> <span m="4322870">virtually</span> <span m="4324370">peck</span> <span m="4326660">everyone</span> <span m="4327180">else,</span> <span m="4329910">we</span> <span m="4330040">will</span> <span m="4330190">call</span> <span m="4330440">a</span> <span m="4330510">king.</span> <span m="4334100">So</span> <span m="4334300">chicken</span> <span m="4334620">that</span> <span m="4334690">virtual</span> <span m="4335160">pecks</span> <span m="4336230">every</span> <span m="4336490">other</span> <span m="4336740">chicken</span> <span m="4348670">is</span> <span m="4348940">called</span> <span m="4355065">a</span> <span m="4355540">king</span> <span m="4356015">chicken.</span> <span m="4361250">So</span> <span m="4361790">let's</span> <span m="4361990">give</span> <span m="4362130">an</span> <span m="4362210">example</span> <span m="4363330">of</span> <span m="4364020">a</span> <span m="4364260">graph.</span> </p>
<p><span m="4365340">So,</span> <span m="4365510">for</span> <span m="4365650">example,</span> <span m="4366220">suppose</span> <span m="4367270">I</span> <span m="4367560">have</span> <span m="4368100">four</span> <span m="4368870">chickens</span> <span m="4370660">that</span> <span m="4371200">know how</span> <span m="4372970">to</span> <span m="4373120">pick</span> <span m="4373460">one</span> <span m="4373620">another</span> <span m="4374000">in</span> <span m="4374110">this</span> <span m="4374370">order.</span> <span m="4382200">So</span> <span m="4382630">who</span> <span m="4382800">in</span> <span m="4382910">the</span> <span m="4383040">pecking</span> <span m="4383690">party</span> <span m="4384140">here</span> <span m="4384460">is</span> <span m="4384920">going</span> <span m="4385110">to</span> <span m="4385200">be</span> <span m="4385370">king?</span> <span m="4386668">Do you</span> <span m="4387162">see</span> <span m="4390130">some</span> <span m="4390400">solutions</span> <span m="4390990">here?</span> <span m="4394740">So</span> <span m="4394930">take,</span> <span m="4395270">for</span> <span m="4395370">example,</span> <span m="4395580">this</span> <span m="4395780">one.</span> <span m="4396760">So</span> <span m="4397760">this</span> <span m="4397990">one</span> <span m="4398305">pecks</span> <span m="4398620">this</span> <span m="4398850">one.</span> <span m="4402830">Because</span> <span m="4403060">we</span> <span m="4403170">talk</span> <span m="4403370">about</span> <span m="4403570">virtually</span> <span m="4403960">pecking,</span> <span m="4404890">it can</span> <span m="4405120">also</span> <span m="4405380">peck</span> <span m="4405870">this</span> <span m="4406080">one</span> <span m="4406220">over</span> <span m="4406500">here.</span> <span m="4407390">It</span> <span m="4407520">does,</span> <span m="4407730">right?</span> <span m="4408780">It</span> <span m="4409020">pecks</span> <span m="4409240">this</span> <span m="4409410">one</span> <span m="4409540">and</span> <span m="4409600">this</span> <span m="4409750">one</span> <span m="4409870">helps</span> <span m="4410230">out,</span> <span m="4411370">and</span> <span m="4411580">can</span> <span m="4411760">peck</span> <span m="4412000">both</span> <span m="4412250">this</span> <span m="4412440">one</span> <span m="4412630">and</span> <span m="4412730">this</span> <span m="4412950">one.</span> <span m="4413590">That's</span> <span m="4413670">cool.</span> <span m="4414610">So</span> <span m="4415220">this</span> <span m="4415420">one</span> <span m="4415580">is</span> <span m="4415740">king.</span> </p>
<p><span m="4417480">And</span> <span m="4417890">this</span> <span m="4418120">one,</span> <span m="4418310">actually--</span> <span m="4420040">let's</span> <span m="4420360">have</span> <span m="4420540">a</span> <span m="4420600">look.</span> <span m="4421650">It</span> <span m="4421950">pecks</span> <span m="4422290">this</span> <span m="4422530">one</span> <span m="4422750">this</span> <span m="4422990">one,</span> <span m="4423860">and</span> <span m="4423970">it</span> <span m="4424110">virtually</span> <span m="4424490">also</span> <span m="4424830">pecked</span> <span m="4425070">this</span> <span m="4425320">one.</span> <span m="4425635">Yay.</span> <span m="4425950">He has</span> <span m="4427090">a</span> <span m="4427140">friend</span> <span m="4427440">over</span> <span m="4427670">here</span> <span m="4427900">that is</span> <span m="4428220">doing</span> <span m="4428450">that</span> <span m="4428610">for</span> <span m="4428770">him.</span> <span m="4429640">So</span> <span m="4429730">this</span> <span m="4430010">one</span> <span m="4430160">is</span> <span m="4430310">actually</span> <span m="4430580">also</span> <span m="4430830">a king.</span> <span m="4431730">So</span> <span m="4432170">you</span> <span m="4432450">can</span> <span m="4432590">have</span> <span m="4432710">multiple</span> <span m="4433140">king</span> <span m="4433440">chickens</span> <span m="4433770">in here.</span> <span m="4435740">What</span> <span m="4435920">about</span> <span m="4436150">this</span> <span m="4436380">one?</span> <span m="4436820">The</span> <span m="4436940">same</span> <span m="4437210">story--</span> <span m="4438960">pecks</span> <span m="4439180">this</span> <span m="4439410">one,</span> <span m="4440540">pecks</span> <span m="4440740">this</span> <span m="4440970">one,</span> <span m="4445550">and</span> <span m="4446000">virtually--</span> <span m="4447020">wait a</span> <span m="4447220">minute.</span> <span m="4452780">The</span> <span m="4453010">one</span> <span m="4453200">on</span> <span m="4453340">the</span> <span m="4453430">left--</span> <span m="4455500">oh yeah,</span> <span m="4455750">over</span> <span m="4455900">here.</span> <span m="4456400">This,</span> <span m="4456785">this.</span> <span m="4458170">So</span> <span m="4458220">this one</span> <span m="4458400">is</span> <span m="4458550">king as</span> <span m="4459040">well.</span> </p>
<p><span m="4460480">Now</span> <span m="4460600">what</span> <span m="4460720">about</span> <span m="4460970">this</span> <span m="4461200">one?</span> <span m="4462510">Well,</span> <span m="4463140">it</span> <span m="4463300">can</span> <span m="4463530">peck</span> <span m="4463720">this</span> <span m="4464050">one.</span> <span m="4465570">And</span> <span m="4465790">then</span> <span m="4466180">in</span> <span m="4466360">one</span> <span m="4466570">more</span> <span m="4466800">step</span> <span m="4467230">from</span> <span m="4467490">here,</span> <span m="4468050">because</span> <span m="4468310">there's</span> <span m="4468470">only</span> <span m="4468720">one</span> <span m="4468920">outgoing</span> <span m="4469410">edge,</span> <span m="4470280">it</span> <span m="4470420">can</span> <span m="4470550">virtually</span> <span m="4471070">peck</span> <span m="4471300">this</span> <span m="4471580">one,</span> <span m="4472580">but</span> <span m="4472730">not</span> <span m="4472910">this</span> <span m="4473160">one.</span> <span m="4473360">So</span> <span m="4473520">this</span> <span m="4473730">one</span> <span m="4473850">is</span> <span m="4473940">definitely</span> <span m="4474310">the loser</span> <span m="4474760">of</span> <span m="4475200">the</span> <span m="4475460">four.</span> <span m="4476740">So</span> <span m="4479010">he's</span> <span m="4479300">not</span> <span m="4479460">the</span> <span m="4479550">king.</span> </p>
<p><span m="4481610">So</span> <span m="4481800">now</span> <span m="4482010">what</span> <span m="4482180">we</span> <span m="4482280">want</span> <span m="4482460">to</span> <span m="4482570">prove</span> <span m="4483030">this</span> <span m="4483880">a</span> <span m="4483970">theorem</span> <span m="4486200">sort</span> <span m="4486440">of</span> <span m="4486640">trans-identify</span> <span m="4488080">one</span> <span m="4488400">of</span> <span m="4488520">the</span> <span m="4488650">chickens</span> <span m="4489370">that we know</span> <span m="4489540">for</span> <span m="4489700">sure</span> <span m="4490130">is</span> <span m="4490260">going to</span> <span m="4490410">be king.</span> <span m="4490970">So</span> <span m="4490990">you</span> <span m="4491100">can</span> <span m="4491240">have</span> <span m="4491410">multiple</span> <span m="4491850">kings.</span> <span m="4492950">But</span> <span m="4493090">maybe</span> <span m="4493340">there's</span> <span m="4493630">one</span> <span m="4493870">chicken</span> <span m="4494510">that,</span> <span m="4494780">from</span> <span m="4495010">our</span> <span m="4495250">intuition,</span> <span m="4496060">we</span> <span m="4496270">may</span> <span m="4496540">feel</span> <span m="4497260">is</span> <span m="4497580">definitely</span> <span m="4498000">going</span> <span m="4498310">to</span> <span m="4498450">be</span> <span m="4499730">king.</span> <span m="4500890">So</span> <span m="4501070">what</span> <span m="4501210">will</span> <span m="4501320">be</span> <span m="4501500">a</span> <span m="4501540">good</span> <span m="4502760">intuition?</span> <span m="4503420">So</span> <span m="4503590">we're</span> <span m="4503680">talking</span> <span m="4504030">here</span> <span m="4504280">about</span> <span m="4504570">virtual</span> <span m="4504950">pecking.</span> </p>
<p><span m="4506130">We</span> <span m="4506310">have</span> <span m="4506800">this</span> <span m="4507000">definition.</span> <span m="4507670">So</span> <span m="4509180">what</span> <span m="4509400">kind</span> <span m="4509650">of</span> <span m="4511040">node</span> <span m="4511450">in</span> <span m="4512620">a</span> <span m="4512670">tournament</span> <span m="4513090">graph</span> <span m="4513560">essentially</span> <span m="4514140">would</span> <span m="4514390">be--</span> <span m="4518836">can</span> <span m="4519280">we</span> <span m="4519420">know</span> <span m="4519590">for</span> <span m="4519720">sure</span> <span m="4520640">that</span> <span m="4521380">it's a</span> <span m="4521650">king?</span> <span m="4521760">Do</span> <span m="4521940">you</span> <span m="4522060">have</span> <span m="4522170">an</span> <span m="4522230">intuition</span> <span m="4522720">for a</span> <span m="4522910">theorem</span> <span m="4523310">that</span> <span m="4523480">you may</span> <span m="4523710">want</span> <span m="4523890">to</span> <span m="4523980">prove?</span> <span m="4525830">So</span> <span m="4526860">that's</span> <span m="4527060">often</span> <span m="4527340">what</span> <span m="4527470">we</span> <span m="4527590">do</span> <span m="4527770">in</span> <span m="4527810">mathematics.</span> <span m="4528460">We have</span> <span m="4528590">some</span> <span m="4528780">kind</span> <span m="4529020">of</span> <span m="4529130">funny</span> <span m="4529460">new</span> <span m="4529710">structure,</span> <span m="4531080">and</span> <span m="4531290">then</span> <span m="4531390">we</span> <span m="4531510">want</span> <span m="4531670">to</span> <span m="4531780">find</span> <span m="4532100">out</span> <span m="4532430">whether</span> <span m="4532890">we can prove</span> <span m="4533230">interesting</span> <span m="4533670">properties</span> <span m="4534140">about</span> <span m="4534430">it.</span> <span m="4534490">So</span> <span m="4534600">we</span> <span m="4534690">start</span> <span m="4535040">to</span> <span m="4535130">search</span> <span m="4535660">for</span> <span m="4535840">actual</span> <span m="4537740">nice</span> <span m="4538080">properties</span> <span m="4538560">and</span> <span m="4538670">theorems.</span> </p>
<p><span m="4539180">So</span> <span m="4539320">in</span> <span m="4539400">this</span> <span m="4539620">case,</span> <span m="4541200">it</span> <span m="4541410">makes</span> <span m="4541710">sense</span> <span m="4542340">that</span> <span m="4543810">the</span> <span m="4544220">vertex that</span> <span m="4544480">has</span> <span m="4544730">the</span> <span m="4544820">most</span> <span m="4545190">outgoing</span> <span m="4545830">edges</span> <span m="4546980">may</span> <span m="4547930">be</span> <span m="4548360">always</span> <span m="4548820">king.</span> <span m="4549490">Can</span> <span m="4549700">we</span> <span m="4550340">prove</span> <span m="4550640">this?</span> <span m="4551820">So</span> <span m="4551980">that's</span> <span m="4552180">what</span> <span m="4552370">we're</span> <span m="4552470">going</span> <span m="4552690">to</span> <span m="4552810">do.</span> <span m="4553920">So</span> <span m="4554080">that's</span> <span m="4554280">the</span> <span m="4554360">theorem.</span> <span m="4555950">And</span> <span m="4556080">let's</span> <span m="4556280">see</span> <span m="4556460">whether we</span> <span m="4556740">can</span> <span m="4558390">do</span> <span m="4558510">this</span> <span m="4559710">in an</span> <span m="4559920">elegant</span> <span m="4560330">way.</span> </p>
<p><span m="4563050">So</span> <span m="4563210">the</span> <span m="4563310">theorem</span> <span m="4563770">is</span> <span m="4564950">that</span> <span m="4565440">even</span> <span m="4565720">though</span> <span m="4565850">there</span> <span m="4566030">are</span> <span m="4566060">multiple</span> <span m="4566560">kings--</span> <span m="4566840">as</span> <span m="4567390">indicated</span> <span m="4567730">in</span> <span m="4568070">that</span> <span m="4568270">particular</span> <span m="4568700">example,</span> <span m="4569270">that</span> <span m="4569410">may</span> <span m="4569580">happen--</span> <span m="4569970">but</span> <span m="4570540">I</span> <span m="4570680">certainly</span> <span m="4571240">know</span> <span m="4572130">that</span> <span m="4572370">the</span> <span m="4572480">chicken</span> <span m="4573480">that</span> <span m="4574320">has</span> <span m="4577870">the</span> <span m="4577960">highest</span> <span m="4579592">outdegree</span> <span m="4583100">is</span> <span m="4583330">definitely</span> <span m="4583570">a</span> <span m="4584038">king.</span> <span m="4590590">And</span> <span m="4591300">the</span> <span m="4591410">way</span> <span m="4591580">we're</span> <span m="4591720">going</span> <span m="4591880">to</span> <span m="4592060">prove</span> <span m="4592390">this</span> <span m="4594140">is</span> <span m="4594960">by</span> <span m="4595150">contradiction.</span> <span m="4595820">Let's</span> <span m="4595980">assume</span> <span m="4596320">that's</span> <span m="4596630">not</span> <span m="4596840">the</span> <span m="4596940">case.</span> <span m="4598080">That</span> <span m="4598270">must</span> <span m="4598490">be</span> <span m="4598630">really,</span> <span m="4599240">really</span> <span m="4599520">weird.</span> </p>
<p><span m="4599860">If</span> <span m="4599990">you</span> <span m="4601240">are</span> <span m="4601440">the</span> <span m="4601560">one</span> <span m="4601800">who</span> <span m="4601990">has</span> <span m="4602400">the</span> <span m="4602530">largest</span> <span m="4602920">outdegree,</span> <span m="4603140">it</span> <span m="4603500">means</span> <span m="4603710">that</span> <span m="4603880">you</span> <span m="4603980">can</span> <span m="4604150">directly,</span> <span m="4605160">just</span> <span m="4605410">by</span> <span m="4605560">yourself,</span> <span m="4606130">peck</span> <span m="4606370">the</span> <span m="4606460">most</span> <span m="4606790">others.</span> <span m="4607760">So</span> <span m="4609010">suppose</span> <span m="4609470">you're</span> <span m="4609670">not</span> <span m="4609790">king.</span> <span m="4613272">So by</span> <span m="4613755">contradiction,</span> <span m="4621630">first</span> <span m="4621950">of</span> <span m="4622000">all,</span> <span m="4623440">let</span> <span m="4624210">U</span> <span m="4625640">have</span> <span m="4626040">the</span> <span m="4626120">highest</span> <span m="4626385">outdegree.</span> <span m="4630910">And</span> <span m="4631050">we</span> <span m="4631070">want</span> <span m="4631340">to</span> <span m="4632390">show</span> <span m="4634630">that this</span> <span m="4634760">U</span> <span m="4634910">is</span> <span m="4635010">king.</span> <span m="4636010">So</span> <span m="4636570">let's</span> <span m="4637190">assume</span> <span m="4637520">the</span> <span m="4637630">contrary.</span> <span m="4639250">So</span> <span m="4639760">let's</span> <span m="4640020">suppose</span> <span m="4643130">that</span> <span m="4643290">U</span> <span m="4643620">is</span> <span m="4643950">not</span> <span m="4644200">king.</span> <span m="4649810">So</span> <span m="4650040">what</span> <span m="4650110">does</span> <span m="4650340">that</span> <span m="4650470">mean?</span> <span m="4651960">So</span> <span m="4652280">let's</span> <span m="4653630">look</span> <span m="4653860">at</span> <span m="4653930">a</span> <span m="4654020">definition</span> <span m="4654710">over</span> <span m="4654960">there</span> <span m="4655930">and</span> <span m="4656130">see</span> <span m="4656380">what</span> <span m="4656530">it</span> <span m="4656670">means</span> <span m="4658870">that</span> <span m="4660580">U</span> <span m="4660810">is not</span> <span m="4661300">king.</span> </p>
<p><span m="4664730">So</span> <span m="4665850">that</span> <span m="4665990">means</span> <span m="4666270">that</span> <span m="4668130">both</span> <span m="4668560">those</span> <span m="4668860">conditions</span> <span m="4669410">are</span> <span m="4669480">violated</span> <span m="4670470">because</span> <span m="4670730">if</span> <span m="4670860">one</span> <span m="4671090">of</span> <span m="4671200">those</span> <span m="4671470">two</span> <span m="4671660">holds,</span> <span m="4675390">I</span> <span m="4675440">know</span> <span m="4675650">there</span> <span m="4675740">must</span> <span m="4676130">be</span> <span m="4676490">one</span> <span m="4676840">vertex,</span> <span m="4677190">V,</span> <span m="4677880">such</span> <span m="4678160">that</span> <span m="4678440">U does</span> <span m="4678940">not</span> <span m="4679250">virtually</span> <span m="4679830">peck</span> <span m="4680400">V.</span> <span m="4681170">So</span> <span m="4681320">I</span> <span m="4681400">know</span> <span m="4681620">that.</span> <span m="4683050">So</span> <span m="4685110">let's</span> <span m="4685350">see</span> <span m="4685480">what</span> <span m="4685640">that</span> <span m="4687520">implies.</span> <span m="4691020">So</span> <span m="4691230">I</span> <span m="4691310">know</span> <span m="4691510">that</span> <span m="4691660">there</span> <span m="4691860">must</span> <span m="4692240">be</span> <span m="4692680">a</span> <span m="4693120">V</span> <span m="4693560">such that</span> <span m="4693770">U</span> <span m="4694830">does</span> <span m="4695060">not</span> <span m="4695250">virtually</span> <span m="4695840">peck</span> <span m="4696570">V.</span> <span m="4699530">So</span> <span m="4700960">maybe</span> <span m="4701230">can</span> <span m="4701370">help</span> <span m="4701570">me</span> <span m="4701750">out.</span> <span m="4702060">What</span> <span m="4702210">does</span> <span m="4702280">that</span> <span m="4702510">mean?</span> <span m="4703900">It</span> <span m="4703980">means</span> <span m="4704280">that</span> <span m="4705040">both</span> <span m="4705410">these</span> <span m="4705600">conditions</span> <span m="4706150">are</span> <span m="4706220">not</span> <span m="4706470">true.</span> </p>
<p><span m="4706950">So</span> <span m="4707000">let's</span> <span m="4707200">look</span> <span m="4707370">at</span> <span m="4707440">the</span> <span m="4707520">first.</span> <span m="4707900">So</span> <span m="4708050">U</span> <span m="4708830">pecks</span> <span m="4709080">V.</span> <span m="4709690">If</span> <span m="4709860">that's</span> <span m="4710100">not</span> <span m="4710310">true,</span> <span m="4711320">and</span> <span m="4711450">we're</span> <span m="4711590">in</span> <span m="4711760">the</span> <span m="4711890">tournament</span> <span m="4712160">graph,</span> <span m="4713350">we</span> <span m="4713500">know</span> <span m="4713770">that</span> <span m="4714230">V</span> <span m="4714540">must</span> <span m="4716770">peck</span> <span m="4717320">U.</span> <span m="4718890">So</span> <span m="4719110">we</span> <span m="4719220">have</span> <span m="4719420">this.</span> <span m="4720350">and</span> <span m="4721010">we</span> <span m="4721220">also</span> <span m="4721560">know</span> <span m="4721790">that</span> <span m="4721990">the</span> <span m="4722140">other</span> <span m="4722550">condition,</span> <span m="4722950">the</span> <span m="4723020">second</span> <span m="4723360">one</span> <span m="4723560">over</span> <span m="4723800">here,</span> <span m="4724020">does</span> <span m="4724230">not</span> <span m="4724370">hold.</span> </p>
<p><span m="4725580">So</span> <span m="4725650">what's</span> <span m="4725800">the</span> <span m="4725880">negation</span> <span m="4729500">of</span> <span m="4729680">this</span> <span m="4729780">second</span> <span m="4730140">condition?</span> <span m="4731120">Maybe</span> <span m="4731330">you</span> <span m="4731390">can</span> <span m="4731520">help</span> <span m="4731720">me</span> <span m="4731860">out.</span> <span m="4733500">So</span> <span m="4734100">we</span> <span m="4734410">have</span> <span m="4734550">here,</span> <span m="4734740">there</span> <span m="4734980">exists</span> <span m="4736010">a</span> <span m="4736120">W</span> <span m="4736515">such</span> <span m="4736910">that</span> <span m="4737060">U</span> <span m="4737340">pecks</span> <span m="4737720">W</span> <span m="4738200">and</span> <span m="4738340">W</span> <span m="4738700">pecks</span> <span m="4739060">V.</span> <span m="4740030">So</span> <span m="4740180">how</span> <span m="4740310">do</span> <span m="4740400">we</span> <span m="4740510">negate</span> <span m="4740900">that</span> <span m="4741780">logical</span> <span m="4742150">expression?</span> </p>
<p><span m="4746059">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4748930">PROFESSOR: For</span> <span m="4749260">all</span> <span m="4749630">W,</span> <span m="4753250">what's</span> <span m="4753510">over there is</span> <span m="4753770">not</span> <span m="4754020">through</span> <span m="4754240">true. So can we</span> <span m="4754660">formulate</span> <span m="4755260">that</span> <span m="4755440">a</span> <span m="4755470">little</span> <span m="4755710">bit</span> <span m="4755830">better?</span> <span m="4756200">So</span> <span m="4756890">it's</span> <span m="4757090">not</span> <span m="4757280">true</span> <span m="4757910">that</span> <span m="4758100">U</span> <span m="4758580">pecks</span> <span m="4758940">W</span> <span m="4759500">and</span> <span m="4760570">W</span> <span m="4761070">pecks</span> <span m="4761570">V.</span> <span m="4763410">So</span> <span m="4763670">that</span> <span m="4763820">means</span> <span m="4764230">that</span> <span m="4765300">either</span> <span m="4769192">U</span> <span m="4770130">pecks</span> <span m="4771110">W</span> <span m="4771790">is</span> <span m="4771930">not</span> <span m="4772090">true,</span> <span m="4772520">or</span> <span m="4773310">W</span> <span m="4773690">pecks</span> <span m="4774070">V</span> <span m="4774380">is</span> <span m="4774560">not</span> <span m="4774750">true.</span> <span m="4775580">So</span> <span m="4775670">let's</span> <span m="4775880">write</span> <span m="4776070">that</span> <span m="4776260">down.</span> <span m="4780550">Let's</span> <span m="4780710">just</span> <span m="4780880">write</span> <span m="4781070">it</span> <span m="4781140">all</span> <span m="4781440">out.</span> <span m="4781780">So</span> <span m="4781830">not</span> <span m="4783930">U</span> <span m="4784870">pecks</span> <span m="4785380">W</span> <span m="4787220">or</span> <span m="4789040">not</span> <span m="4790740">W</span> <span m="4790920">pecks</span> <span m="4791350">V.</span> <span m="4794170">Well,</span> <span m="4794460">how</span> <span m="4794690">can</span> <span m="4794840">we</span> <span m="4794980">rewrite</span> <span m="4795440">this?</span> <span m="4796520">Well, it's a</span> <span m="4796740">tournament</span> <span m="4797160">graph,</span> <span m="4797810">so</span> <span m="4797990">we</span> <span m="4798110">know</span> <span m="4798330">that</span> <span m="4798950">W</span> <span m="4799430">pecks</span> <span m="4801650">U.</span> <span m="4805520">Or</span> <span m="4807160">this</span> <span m="4807320">particular</span> <span m="4807730">condition,</span> <span m="4808800">which</span> <span m="4809110">is</span> <span m="4810880">V</span> <span m="4811690">pecks</span> <span m="4812300">W.</span> </p>
<p><span m="4814640">So</span> <span m="4814840">what</span> <span m="4814940">do</span> <span m="4815080">I</span> <span m="4815180">have</span> <span m="4815510">here?</span> <span m="4815860">Is</span> <span m="4816020">this</span> <span m="4816200">going</span> <span m="4816620">in</span> <span m="4816720">the</span> <span m="4816820">right</span> <span m="4816910">direction?</span> <span m="4820540">Well,</span> <span m="4821230">I</span> <span m="4821330">want</span> <span m="4821550">to</span> <span m="4821820">prove</span> <span m="4822210">something</span> <span m="4822920">about--</span> <span m="4826500">if</span> <span m="4827020">I</span> <span m="4827070">use</span> <span m="4827340">contradiction</span> <span m="4828780">and</span> <span m="4828970">I</span> <span m="4829020">suppose</span> <span m="4829400">that</span> <span m="4829490">U</span> <span m="4829710">is</span> <span m="4829850">not</span> <span m="4830020">the</span> <span m="4830110">king,</span> <span m="4832110">I've</span> <span m="4832530">assumed</span> <span m="4833070">that</span> <span m="4833220">U</span> <span m="4833460">has</span> <span m="4833600">the</span> <span m="4833660">highest</span> <span m="4833990">outdegree.</span> <span m="4834600">So I</span> <span m="4834720">want</span> <span m="4834870">to</span> <span m="4834960">show</span> <span m="4835270">that</span> <span m="4835510">I</span> <span m="4835580">somehow</span> <span m="4835970">violated.</span> <span m="4836690">So</span> <span m="4836860">somehow</span> <span m="4837280">I'm</span> <span m="4837580">able</span> <span m="4838470">to</span> <span m="4838640">construct</span> <span m="4839600">some</span> <span m="4839880">vertex,</span> <span m="4840210">V.</span> <span m="4841680">And</span> <span m="4842020">by</span> <span m="4842480">negating</span> <span m="4843090">that</span> <span m="4843260">U</span> <span m="4843490">is not a</span> <span m="4843690">king</span> <span m="4844030">it</span> <span m="4844130">seems</span> <span m="4844370">that</span> <span m="4844490">this</span> <span m="4844710">vertex,</span> <span m="4844910">V,</span> <span m="4845790">makes</span> <span m="4846110">a</span> <span m="4846150">really</span> <span m="4846400">good</span> <span m="4846620">candidate</span> <span m="4847880">to</span> <span m="4848010">show</span> <span m="4848350">that</span> <span m="4848480">there's</span> <span m="4848840">a</span> <span m="4848890">higher</span> <span m="4849550">degree</span> <span m="4849800">outdegree than</span> <span m="4850000">U.</span> <span m="4850830">So</span> <span m="4851000">let's</span> <span m="4851210">see</span> <span m="4851390">whether</span> <span m="4851560">we</span> <span m="4851660">can</span> <span m="4851820">do</span> <span m="4852000">this.</span> </p>
<p><span m="4855040">We</span> <span m="4855060">can</span> <span m="4855270">rewrite</span> <span m="4855800">this</span> <span m="4856400">logical</span> <span m="4856790">expression</span> <span m="4857410">once</span> <span m="4857720">more.</span> <span m="4858070">We</span> <span m="4858180">can</span> <span m="4858370">also</span> <span m="4858600">say</span> <span m="4859470">that,</span> <span m="4860010">well,</span> <span m="4860570">if</span> <span m="4862630">U</span> <span m="4862980">pecks</span> <span m="4863330">W--</span> <span m="4865550">so</span> <span m="4865780">this</span> <span m="4865990">is</span> <span m="4866090">not</span> <span m="4866290">true--</span> <span m="4867560">then</span> <span m="4869000">it</span> <span m="4869160">must</span> <span m="4869490">be</span> <span m="4869630">true</span> <span m="4870090">that</span> <span m="4870270">this</span> <span m="4870500">one</span> <span m="4870720">holds</span> <span m="4871860">because</span> <span m="4874800">if</span> <span m="4874920">that's</span> <span m="4875120">not</span> <span m="4875240">the</span> <span m="4875330">case,</span> <span m="4875700">then</span> <span m="4875830">this</span> <span m="4876000">condition</span> <span m="4876420">is</span> <span m="4876560">not</span> <span m="4876730">true.</span> <span m="4877720">So</span> <span m="4877900">if</span> <span m="4878080">U</span> <span m="4878490">pecks</span> <span m="4878990">W,</span> <span m="4880260">then</span> <span m="4880450">this</span> <span m="4880650">is</span> <span m="4880780">not</span> <span m="4881010">true.</span> <span m="4881280">So then</span> <span m="4881550">it</span> <span m="4881670">must</span> <span m="4881920">be</span> <span m="4881990">the</span> <span m="4882120">case</span> <span m="4882410">that</span> <span m="4882640">that</span> <span m="4882790">is true.</span> <span m="4883200">So</span> <span m="4883520">V</span> <span m="4885330">pecks</span> <span m="4885610">W.</span> <span m="4885910">So</span> <span m="4886200">now</span> <span m="4886720">let's</span> <span m="4886920">have</span> <span m="4887120">a</span> <span m="4887190">look</span> <span m="4887420">at</span> <span m="4887600">V.</span> <span m="4888960">We</span> <span m="4889100">noticed</span> <span m="4889700">that</span> <span m="4890040">for</span> <span m="4890370">all</span> <span m="4890730">outgoing</span> <span m="4891360">edges</span> <span m="4891710">from</span> <span m="4891910">U,</span> <span m="4895130">there</span> <span m="4895710">exists</span> <span m="4895830">a</span> <span m="4895880">similar</span> <span m="4896270">outgoing</span> <span m="4896830">edge</span> <span m="4897460">for</span> <span m="4897770">V.</span> <span m="4899890">But</span> <span m="4900090">V</span> <span m="4900320">has</span> <span m="4900550">one</span> <span m="4900900">more</span> <span m="4901180">outgoing</span> <span m="4901820">edge.</span> <span m="4902570">V,</span> <span m="4902840">actually, is an</span> <span m="4903270">outgoing</span> <span m="4903720">edge</span> <span m="4903930">to</span> <span m="4904210">U.</span> </p>
<p><span m="4905030">So</span> <span m="4905260">what do</span> <span m="4905410">we</span> <span m="4905510">see</span> <span m="4905840">here?</span> <span m="4906870">That</span> <span m="4907130">the outdegree</span> <span m="4909420">of</span> <span m="4909850">V</span> <span m="4911120">is</span> <span m="4911420">actually</span> <span m="4912710">at</span> <span m="4912890">least</span> <span m="4913520">the</span> <span m="4913690">outdegree</span> <span m="4917180">of</span> <span m="4917400">U--</span> <span m="4918490">which</span> <span m="4918930">is</span> <span m="4919720">this</span> <span m="4919900">particular</span> <span m="4920350">condition</span> <span m="4920840">over</span> <span m="4921100">here</span> <span m="4922040">that we</span> <span m="4922180">show</span> <span m="4922510">here,</span> <span m="4922740">for</span> <span m="4922930">all</span> <span m="4923140">the</span> <span m="4923260">W</span> <span m="4923400">we</span> <span m="4924100">have</span> <span m="4924390">that this</span> <span m="4924690">is</span> <span m="4924820">true--</span> <span m="4925960">plus</span> <span m="4926670">and</span> <span m="4926950">we</span> <span m="4927070">have</span> <span m="4927200">an</span> <span m="4927250">extra</span> <span m="4927620">one.</span> <span m="4928690">It's</span> <span m="4928860">this</span> <span m="4929080">one.</span> <span m="4929990">Oh, but</span> <span m="4930260">now</span> <span m="4930490">we</span> <span m="4930600">have</span> <span m="4930750">a</span> <span m="4930810">contradiction</span> <span m="4931500">because</span> <span m="4931900">we</span> <span m="4931980">said</span> <span m="4932260">that</span> <span m="4932350">U</span> <span m="4932615">has</span> <span m="4932880">the</span> <span m="4932960">highest</span> <span m="4933510">degree.</span> <span m="4934830">But</span> <span m="4934970">it</span> <span m="4935080">turns</span> <span m="4935430">out</span> <span m="4935630">that</span> <span m="4935730">you</span> <span m="4935880">have</span> <span m="4936050">constructed</span> <span m="4936550">one</span> <span m="4936750">that</span> <span m="4936840">has</span> <span m="4937060">a</span> <span m="4937100">higher</span> <span m="4937560">outdegree.</span> <span m="4938750">So</span> <span m="4938900">that's</span> <span m="4939090">a</span> <span m="4939180">contradiction.</span> <span m="4940810">That means</span> <span m="4941110">that</span> <span m="4941230">our</span> <span m="4941610">original</span> <span m="4941970">assumption</span> <span m="4944150">is</span> <span m="4944360">actually</span> <span m="4944840">wrong.</span> <span m="4945260">So</span> <span m="4945910">suppose</span> <span m="4946440">that</span> <span m="4946790">U is not the</span> <span m="4946950">king</span> <span m="4947330">was</span> <span m="4947520">a</span> <span m="4947610">wrong</span> <span m="4947860">assumption,</span> <span m="4949010">and</span> <span m="4949190">U</span> <span m="4949370">must</span> <span m="4949610">be</span> <span m="4949750">king.</span> <span m="4951480">So</span> <span m="4951650">that's</span> <span m="4951850">the</span> <span m="4951950">end</span> <span m="4952090">of</span> <span m="4952160">this</span> <span m="4952370">proof.</span> </p>
<p><span m="4956290">This</span> <span m="4956470">is</span> <span m="4956610">the</span> <span m="4956750">end</span> <span m="4956930">of</span> <span m="4957060">this</span> <span m="4957230">lecture.</span> <span m="4958780">So</span> <span m="4959000">see you</span> <span m="4959240">tomorrow</span> <span m="4959610">at</span> <span m="4960020">recitation</span> <span m="4960710">and</span> <span m="4960990">next week</span> <span m="4961410">we will</span> <span m="4961720">continue with</span> <span m="4962030">communication</span> <span m="4962700">graphs</span> <span m="4963540">and</span> <span m="4963930">partial</span> <span m="4964420">orderings.</span> </p>
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